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The Zel'dovich Electric Dipole Moment in Atoms

D. E. Krause

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

It is generally believed that both parity (P) and time reversal (T) invariance must be broken to produce a permanent electric dipole moment in a stable non-degenerate rotationally invariant system. However, Zel'dovich proposed that the P-violating weak interaction alone might be sufficient to produce an electric dipole moment in an unstable system, where the exponential decay law produces an "arrow of time" without explicit T-violation in the Hamiltonian. This "Zel'dovich moment" would be proportional to the linewidth of the unstable system. Subsequently, Bernreuther and Nachtmann explicitly calculated this Zel'dovich moment for deuterium. Here we use the master equation formalism to study the problem of the Zel'dovich moment. We apply a widely used master equation to a 3-level atom to obtain the results of Bernreuther and Nachtmann, and show that their effect may persist in steady-state situations. We then use a similar equation to study a 2-level atom, where we find an even more dramatic result. If true, it indicates that all atoms could have a Zel'dovich moment, even when they are in their ground states. We then show that these latter results are spurious--they arise because the master equation used neglected terms of the same order of magnitude as the Zel'dovich moment. When included, the Zel'dovich moment disappears in the 2-level atom. Since Bernreuther and Nachtmann's result was obtained in a similar manner, it is also called into question.

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Quantum Rabi Model Beyond the Electric Dipole Approximation

S. Zarrinkamar

We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].

•Oct 28, 2026•0 citations

Quantum Rabi Model Beyond the Electric Dipole Approximation

S. Zarrinkamar

We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].

•Oct 28, 2026•0 citations
It is generally believed that both parity (P) and time reversal (T) invariance must be broken to produce a permanent electric dipole moment in a stable non-degenerate rotationally invariant system. However, Zel'dovich proposed that the P-violating weak interaction alone might be sufficient to produce an electric dipole moment in an unstable system, where the exponential decay law produces an "arrow of time" without explicit T-violation in the Hamiltonian. This "Zel'dovich moment" would be proportional to the linewidth of the unstable system. Subsequently, Bernreuther and Nachtmann explicitly calculated this Zel'dovich moment for deuterium. Here we use the master equation formalism to study the problem of the Zel'dovich moment. We apply a widely used master equation to a 3-level atom to obtain the results of Bernreuther and Nachtmann, and show that their effect may persist in steady-state situations. We then use a similar equation to study a 2-level atom, where we find an even more dramatic result. If true, it indicates that all atoms could have a Zel'dovich moment, even when they are in their ground states. We then show that these latter results are spurious--they arise because the master equation used neglected terms of the same order of magnitude as the Zel'dovich moment. When included, the Zel'dovich moment disappears in the 2-level atom. Since Bernreuther and Nachtmann's result was obtained in a similar manner, it is also called into question.
The effect of an electric quadrupole moment QQQ is studied for positron-atom bound systems. It is demonstrated that for Q>50Q>50Q>50 a.u. the electric quadrupole potential is sufficiently strong to bind a positron (or an electron) even in the absence of the dipole polarization potential. Such large values of QQQ are not known for atomic ground states; however, they exist in molecules and excited atoms. In the 2s2p3P2o2s2{p}^{3}{P}_{2}^{o}2s2p3P2o​ state of beryllium, the quadrupole contribution makes the difference between a stable bound state and an unstable state, which decays to a Be+\mathrm{Be}{}^{+}Be+ ion and positronium. In the majority of atoms the quadrupole contribution is small and can be neglected.
Permanent electric dipole moments in molecules require a breaking of parity symmetry. Conventionally, this symmetry breaking relies on the presence of heteronuclear constituents. We report the observation of a permanent electric dipole moment in a homonuclear molecule in which the binding is based on asymmetric electronic excitation between the atoms. These exotic molecules consist of a ground-state rubidium (Rb) atom bound inside a second Rb atom electronically excited to a high-lying Rydberg state. Detailed calculations predict appreciable dipole moments on the order of 1 Debye, in excellent agreement with the observations.
The Rabi Hamiltonian, describing the interaction between a two-level atomic system and a single-cavity mode of the electromagnetic field, is one of the fundamental models in quantum optics. The model becomes exactly solvable by considering an atom without permanent dipole moments, whose excitation energy is quasiresonant with the cavity photon energy, and by neglecting the nonresonant (counter-rotating) terms. In this case, after including the decay of either the atom or the cavity mode to a continuum, one can derive the well-known phenomenology of quasiresonant transitions, including the fluorescence triplets. In this work we consider the most general Rabi model, incorporating the effects of permanent atomic electric dipole moments. Based on a perturbative analysis, we compare the intensities of emission lines induced by rotating terms, counter-rotating terms, and parity-symmetry-breaking terms in order to identify the parameter regimes in which these different contributions play a significant role. The analysis reveals that the emission strength related to the existence of permanent dipoles may surpass the one due to the counter-rotating interaction terms but is usually much weaker than the emission due to the main, resonant coupling. This ratio can be modified in systems with a reduced dimensionality or by engineering the energy spectral density of the continuum.
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].