Quantum Field Theory for Multipolar Composite Bosons with Mass Defect and Relativistic Corrections

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.

Popular Summary

Atomic sensors are powerful devices for testing general relativity and the standard model. To fully harness the advantage of their quantum nature, including their motion, internal structure, and many-body behavior, an effective field theory is the method of choice. We derive such a description from the underlying fundamental theory. Our results open the door for clocks and matter-wave interferometers detecting relativistic effects with a precision higher than allowed by classical physics. At the same time, we find the mass defect originating from the energy-mass equivalence, where the internal energies of the atom lead to different masses that influence the atom’s motion. This effect constitutes the basis for quantum clock interferometry.

We follow a top-down approach, starting with quantum electrodynamics to describe bound systems at low energies, leading eventually to ultracold atoms. Our theory treats interacting ensembles of spin carrying, possibly charged composite bosons, such as atoms or ions. Based on the motion of the constituents of each atom and its relativistic corrections, we derive the atomic energy structure, its effect on the motion of an atom, and scattering potentials between atoms, as well as the interaction with light. With that, we find corrections to the Gross-Pitaevskii equation, including the mass defect, a nonlinear Schrödinger-type equation that describes degenerate quantum gases like Bose-Einstein condensates.

Our method can be generalized to effective field theories on other length and energy scales, such as multispecies quantum gases or cold molecules. The results can directly be applied to modeling atomic sensors for dark matter, the theory of relativity, or gravitational waves.

Figure 1

Hierarchy of effective field theories: At the top level, quantum electrodynamics (QED) describes the interaction of fermions (dots), in particular electrons (blue) and positrons (black), with photons, where real (measurable) photons are represented by zigzag lines and virtual photons between two fermions by wiggly lines. The respective field operators are the four-component spinors $\hat{\psi }$ and $\hat{\overline{\psi }}$ and four potential ${\hat{A}}^{\mu }$, whose interaction is described by typical Feynman diagrams, such as the vacuum polarization (solid lines with arrows are fermions) shown on the right. The next refined field theory is nonrelativistic QED (NRQED), where the antiparticle can be removed from the description by the restriction to an NR limit of QED. In NRQED fermion field operators are replaced by two-component spinors ${\hat{\psi }}_i$ creating and annihilating electrons or nuclei, where the latter are assumed as elementary fermions as well. These fermions can interact with each other. For example, the Feynman diagram shows two solid lines representing now two fermions scattering via a virtual photon. In a next step potential NRQED (pNRQED) replaces virtual photons in a second-order scattering process by a nonlocal potential ${\hat{V}}^{(ij)}$ for an effective first-order scattering process, where external photons are completely described by the vector potential $\hat{\mathbf{A}}$ in Coulomb gauge. This level is the starting point of our work. The cobosonic subspace follows from a projection, where we restrict the Hilbert space to only electron-nucleus pairs (cobosons) and introduce a separation of scales, i.e., atomic dimensions and scattering length. As a consequence, fermionic field operators are replaced by cobosonic field operators $\hat{\phi }$ and interactions between cobosons (orange) are mediated by scattering potentials ${\hat{V}}_{\mathrm{scatt}}$. Here, Feynman diagrams now feature double solid lines representing a coboson. Multipolar cobosonic subspace describes external photons by EM fields $\hat{\mathbf{E}}$ and $\hat{\mathbf{B}}$ (dashed zigzag line) interacting with cobosons via multipoles whose degrees of freedom are described by c.m. and relative coordinates.

Figure 2

Relevant Feynman diagrams up to order $c^{-2}$. The dashed and wiggly lines correspond to scalar photons resulting from a contraction of the scalar potential and vector photons arising from contracting vector potential components, respectively. The vertices are labeled according to the specific term in the Hamiltonian density being contracted: electric energy (red), kinetic energy (blue), spin-magnetic field (green, ${\overline{c}}_{\mathrm{F}}$), Darwin term (purple, ${\overline{c}}_{\mathrm{D}}$), spin-orbit term (orange, ${\overline{c}}_{\mathrm{S}}$), and contact interaction (brown, $d_1$ and $d_2$). Details regarding $c$- and $d$-type Wilson coefficients can be found in Appendix pp1. The potential connected to each Feynman diagram is calculated in both Coulomb (top, blue shaded) and Lorenz gauge (bottom, orange shaded). The Feynman diagrams of the first row yield different potentials that depend on the gauge, whereas the potentials of the second row are identical in both gauges. The overall effective potential is the sum of all contributions from the first and second row and is gauge invariant. The potential is given in units of ${\kappa }_{ij}=-q_i{q}_j/(8\pi {\epsilon }_0{m}_i{m}_j{c}^2)$ and we introduce abbreviations for individual minimally coupled angular momenta ${\widehat{\overline{\mathit{\ell }}}}_i=\mathbf{r}\times {\widehat{\overline{\mathbf{p}}}}_i$ and relative distance $\mathbf{r}={\mathbf{x}}_i-{\mathbf{x}}_j^{\mathrm{\prime }}$ omitting indices $i$ and $j$ for simplicity. In addition, we defined the spin contribution ${\hat{S}}_{ij}=-{\hat{\mathbf{s}}}_i\cdot {\hat{\mathbf{s}}}_j^{\mathrm{\prime }}+3(\mathbf{r}\cdot {\hat{\mathbf{s}}}_i)(\mathbf{r}\cdot {\hat{\mathbf{s}}}_j^{\mathrm{\prime }})/|\mathbf{r}{|}^2$ of the magnetic dipole-dipole potential.

Figure 3

(a) The left-hand side shows the situation in pNRQED, where all elementary fermions of the system interact with each other and no interaction is dominant compared to the others. A projection to the coboson subspace introduces length scales, and by that our system is only composed of uniquely assigned electron-nucleus pairs. In particular, we find an atomic length scale $a$ associated with the inner-atomic distance between the constituents and a scattering length scale $b$ associated with the distance between different cobosons. The separation of scales allows for the identification of the dominant binding potential ${\hat{V}}_{\mathrm{b}}={\hat{V}}^{(ne)}+{\hat{V}}^{(en)}$ between the constituents of a coboson compared to the weaker scattering potential ${\hat{V}}_{\mathrm{scatt}}={\hat{V}}^{(nn)}+{\hat{V}}^{(ne)}+{\hat{V}}^{(en)}+{\hat{V}}^{(ee)}$ composed of attractive and repulsive interactions among constituents of different cobosons. (b) The separation of scales has also implications for the allowed position of different cobosons. The nucleus of coboson 1 can be placed anywhere, but the corresponding electron only within a vicinity of radius $a$ around it. The nucleus of coboson 2 must not be closer than a distance $b$ from the nucleus of coboson 1. As a consequence, the purple sphere around coboson 1 is excluded for coboson 2. Similarly, coboson 3 must not be located in the purple and green spheres around cobosons 1 and 2, respectively.

Figure 4

Due to the mass defect, the energy-momentum dispersion of a neutral coboson (with $Q=0$) depends on the unperturbed internal state defined by the principal quantum number $n=1,2,3,\dots$ . Additional relativistic corrections cause a splitting into sublevels that correspond to the fine and hyperfine structure given by the set of quantum numbers $(j,S,\ell )$. These structures do not enter the modified energy-momentum dispersion at our level of approximation in $c^{-2}$. The quantum number $j$ coincides with the quantum number $f$ that is the standard form for the hyperfine splitting in atomic physics and results from coupling the electron’s spin with the relative angular momentum, before coupling the nuclear spin.

Figure 5

Form of the scattering potential. (a) Two ionized cobosons with $Q\ne 0$ experience a repulsive Coulomb potential as the dominant contribution of their interaction. (b) For neutral cobosons with $Q=0$ higher-order dipole terms are dominating the interaction. The panel shows the dipole-dipole potential $V_{\text{d-d}}\propto (Z-1{)}^2/|\mathbf{\Delta }\mathbf{R}|+Z{a}^2(1-3{\cos }^2\vartheta )/|\mathbf{\Delta }\mathbf{R}{|}^3$ for the simplified case of $\mathbf{r}={\mathbf{r}}^{\mathrm{\prime }}=a{\mathbf{e}}_r$, where $\vartheta$ is the angle $\mathbf{r}$ and $\mathbf{\Delta }\mathbf{R}$. Depending on the orientation, the potential is either attractive or repulsive.

Figure 6

Sketch of the time evolution of noninteracting cobosons in a superposition of ground $|g⟩$ and excited $|e⟩$ state. The c.m. probability distribution $|{\mathrm{\Psi }}_j(x,t){|}^2$ of the atom in state $|j⟩$ is shown by different wave packets. Their form depends on the internal states due to different dispersion relations, which depend on the masses $M_g$ and $M_e$ due to the mass defect. Moreover, if both wave packets have the same initial momentum, the ground state propagates with a higher velocity.

Figure 7

(a) Set of all possible Feynman diagrams of two real fermions (which model in our treatment the constituents of a composite particle), interacting with EM fields. It can be described by the exchange of purely virtual photons (VP) and real photons (zigzag lines). The former can be any combination of scalar photons (dashed line), and vector photons (wiggly line) mediating the interaction between both fermions. This set of Feynman diagrams can be expanded into an increasing amount of scattering processes with real photons that account for external EM fields. (b) The lowest-order nontrivial set of virtual photons includes all possible Feynman diagrams with one virtual scalar or vector photon (second-order scattering). All diagrams are reduced to an effective first-order scattering by matching to an effective potential ${\hat{V}}^{(ij)}$ in the framework of pNRQED.

Figure 8

Canonical momenta appear in the effective potentials of the orbit-orbit (LL) coupling ${\hat{V}}_{\mathrm{LL}}^{(ij)}$ and the spin-orbit (LS) coupling ${\hat{V}}_{\mathrm{LS}}^{(ij)}$. To replace these by minimally coupled momentum operators, we augment the orbit-orbit Feynman diagram by three additional diagrams: two fermions $i$ and $j$ exchanging a virtual photon while they scatter a real photon, respectively, together with the diagram where simultaneously both fermions scatter a real photon. Both spin-orbit diagrams exchanging a virtual vector or a virtual scalar photon, respectively, have to be augmented by one additional diagram describing the scattering of a real photon during this process.