The research on the interactions between electromagnetic field and matter plays a centre role in the progress of contemporary physics. A fundamental startpoint is to consider the coupling between atom and light, which is responsible for many physical phenomena. For example, the light emission and absorption of atoms and molecules are powerful sources to detect the underlying atomic structures and dynamic processes. It also enables us to manipulate the atomic internal and external degrees of freedom with the atom–light coupling, to realize states near or far from the thermodynamic equilibrium.^[1–4]

The physics of ultracold atoms focuses on coherent quantum motion of atoms in the ultra-low temperature and many-body effects in these quantum gases due to the interaction between atoms.^[5–8] Interaction between atoms and laser lights plays an important role in controlling cold atom systems, ranging from matured technologies such as laser cooling, optical trapping, and optical lattices to the latest developments such as synthetic gauge field and stimulated Raman adiabatic passage to achieve the ground state polar molecules.^[9–13] In most of the studies in cold atom physics, the atom–light interaction is treated in the frame of dipole approximation which refers to the interaction between atomic electric dipole and the laser light, while the electric quadrupole interaction effect is usually neglected. It was shown that the rarely considered atomic electric quadrupole may lead to fractional frequency shifts in an optical lattice clock.^[14–17] Nevertheless, the corrections there are estimated in the harmonic oscillator approximation and the general result brought by the electric quadrupole interaction is not clear. Especially, in what circumstances the electric quadrupole interaction may play a critical role and cannot be neglected is a question. In this paper to address this problem, we go beyond the harmonic approximation and take into account the dipole and quadrupole interactions on an equal foot.

The rest of this paper is organized as follows. In Section 2, we present the general formalism of atom–light coupling, and derive the effective Hamiltonians for the electric dipole and quadrupole interaction in a perturbative way, respectively. Then based on the results, we analyze the ground-state energy corrections for both processes in Section 3. Finally in Section 4, we give a brief summary.

It is well known from a textbook of classical electrodynamics that, if a localized charge distribution described by is placed in an external potential , the electrostatic energy of the system is:^[18]

If the potential Φ is slowly varying over the region where is noneligible, then it can be expanded as a Taylor series around a suitably chosen origin, which is

Utilizing the definition of the electric field and taking , equation (2) can be rewritten as

Substituting into Eq. (1), the energy H becomes

In general, due to the total charge of an atom is zero, the dipole term becomes the most important one in the atom–light interaction. In this paper, without loss of generality, we take an oscillatory light field with vector component E_j and frequency ω, where is the wave vector of the light field and is the initial phase. To facilitate the following discussions, we turn to the quantization description by replacing the observable quantities with the corresponding operators, and write the atom–light coupling Hamiltonian as with

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Typical atomic energy level structure. and are the detuning of the electric dipole transition (red arrow) and the quadrupole transition (blue arrow), respectively. and are the energy of the excited P and D orbital states, respectively. is the energy of the optical field.

The atom is described by the Hamiltonian

In the frame rotating with frequency ω via a unitary transformation , the atomic Hamiltonian transforms as , while the dipole Hamiltonian becomes

Dipole interaction is thoroughly discussed in Ref. [19]. In the regime of where is the detuning of the dipole transition, the effect of the fine-structure splitting can be neglected. In case, the explicit form of above Hamiltonian takes

When the effect of the fine-structure splitting is considered, the dipole interaction yields a vector light shift in addition to the scalar light shift of Eq. (10) as

Let us now turn to the quadrupole transition (see the blue arrow in Fig. 1.), similar treatment can be applied. After a tedious derivation (see Appendix A for detail), the quadrupole Hamiltonian in the rotating frame under RWA becomes

When we first consider no fine structure case with , the corresponding effective perturbation Hamiltonian gives

Let us then consider a finite fine-structure coupling. Using the relation with , the corresponding effective perturbation Hamiltonian gives

We now turn to investigate the effect of the dipole and quadrupole interaction on the ground state energy. It is shown that in Eqs. (11) and (15) that the vector light shift is about times smaller than the scalar light shift for both dipole interaction and quadrupole interaction. To get a glance of the effects of dipole and quadrupole interaction, here we only compare the scalar light shifts due to electric dipole interaction and electric quadrupole interaction. From Eqs. (10) and (13), we can obtain the ground state energy shifts induced by the dipole and quadrupole transitions, which are given by

The energy corrections are then characterized by the ratio between two transitions, which is

Now we do numerical comparisons for alkali atoms as examples to get a more intuitive understanding. It is reasonable to use hydrogen atom wave functions ^[4,25,26] to perform the integration. Here are set to be . The magnitude of electric wave vector and ω are obtained from the energy level where R is the Rydberg constant, h is Planckʼs constant, c is the speed of light, n is the principal quantum number, and is Rydbergʼs correction which describes an angular momentum-dependent quantum defect. Rydbergʼs correction is independent of n, but it is a function of the quantum number l and decreases rapidly with l. Here we consider only the ground state and use the data for from Ref. [4] and [26]. The dipole and quadrupole transition matrix elements, and , and the ratio of two effects on ground state energy are listed in Table 1, η are illustrated in Fig. 2 and are illustrated in Fig. 3.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) The transition matrix element ration η in terms of principle quantum number n. η is in unit of , where a0 is the Bohr radius.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) The comparison of the impacts of the electric dipole and quadrupole coupling terms on the atomic ground state manifold to second order in perturbation theory, , in terms of principle quantum number n.

Table 1.

Table 1.

Table 1. Electric dipole and quadrupole impacts on the alkali atomic ground state manifold. . Lithium Sodium Potassium Rubidium Caesium Francium Atom number 3 11 19 37 55 87 Electron configuration 2S1 3S1 4S1 5S1 6S1 7S1 0.41 1.37 2.23 3.19 4.13 ∼5.10 9.39 29.9 72.5 149.18 274.19 464.53 184.68 2591.71 17496.8 79834.8 283056 839901 /Hz 1010 1010 1010 1010 1010 1010 ω/Hz 8.11×1014 7.63×1014 6.21×1014 5.88×1014 5.41×1014 5.20×1014 3.02×10−7 1.18×106 2.18×106 6.59×106 1.54×105 3.37×105

Table 1. Electric dipole and quadrupole impacts on the alkali atomic ground state manifold. .

Figure 2 shows that the ratio η is increasing with the principle quantum number n which means is increasing much more rapidly than . Figure 3. shows that for all alkali atoms the ground state energy shift by electric quadrupole interaction is about times smaller than that by electric dipole interaction. This behavior can be understood from Eq. (18). Taking Bohr radius and ω about 10¹⁴ Hz, is then about which makes the effect by electric quadrupole interaction much less than that by electric dipole interaction even the quadrupole transition element is times larger than the dipole transition matrix element in units of . Physically, means that the optical wave length is much larger than the size of the atom. And in this regime, the atom sees a spatially almost constant electric field and therefore the quadrupole effect which couples to the electric field gradient can be negligible, suggesting that the electric dipole approximation used in previous literature works well for alkali-like atoms.

Despite of the validity of the dipole approximation for alkali atoms, there may exist the situations where the quadrupole effect is considerable and one needs to go beyond the dipole approximation. First, we notice that, we have used a plane wave light field in above discussions, which naturally falls into the electric dipole approximation regime with . If we consider a short pulse laser with much larger electric field gradient, it may lead to a much larger electric quadrupole contribution. Second, for a Rydberg atom, which is an excited atom with one or more electrons that have a very high principal quantum number, it has a very large atom size which is comparable to the light field wave length. Under this condition, the electric quadrupole interaction might become important and cannot be ignored. Third, in this paper, we mainly consider the alkali-like atoms which have large electric dipole moments. While for the molecule with a relative large quadrupole moment such as the carbon dioxide, the effect of the quadrupole interaction may become significant.

In this paper, we have studied the dipole and quadrupole coupling between an atom and a light field. In the far off-resonance case, we derive the effective Hamiltonians to describe the effects on the ground state manifold brought by the dipole and quadrupole transitions. We find that, for the alkali-like atoms, the energy correction from dipole transition is dominant and the dipole approximation can be applied. While for other cases like the Rydberg atom with a very large size, the molecular with large quadrupole moment, the correction induced by the quadrupole interaction may become considerable and one needs to go beyond the dipole approximation. Future studies would include Rydberg atoms, a gaussian-like light field, a many-electron atom, and so on.