Effect of the Atomic Dipole-Dipole Interaction on the Phase Diagrams of Field-Matter Interactions I: Variational procedure

Recently, we have studied the quantum phase diagrams of a system of $n$-level atoms interacting with $\ell$ electromagnetic modes in a cavity, under the dipolar aproximation Cordero et al. (2021); López-Peña et al. (2021).

When the inter-atomic distance of a cold atomic gas is comparable to the wavelength of the electromagnetic field, the dipole-dipole coupling between the atoms becomes important and yields relevant collective effects Weiner et al. (1999). These Van der Waals forces, due to dipole-dipole interactions of the induced electric dipole moments, become important and must be taken into account. However, one needs to be careful about the long or short character of the dipolar potential for many particle systems, as one can find in theoretical and experimental studies of ultra-cold boson systems Jones et al. (2006); Lahaye et al. (2009); Astrakharchik and Lozovik (2008).

The dipole-dipole interaction decays as $1/r^{3}$, with $r$ the distance between particles, and is thus of a different nature as the matter-field interaction considered in earlier works (cf., e.g., Cordero et al. (2015) and references therein). Energy transfer between the particles (atoms, molecules) is one of the important consequences of this interaction. For Rydberg atoms it is particularly interesting, as they have high principal quantum numbers $n$, while the dipole moment scales as $n^{2}$ in atomic units Gerry and Knight (2005).

A review of theoretical and experimental work on the dipole-dipole interaction between Bose-Einstein condensates has been presented in Lahaye et al. (2009). The trapping of cooled polar molecules and other atomic species Doyle et al. (2004) was important for attracting the attention to study these type of interactions, and the long-range dipole-dipole interaction in low-density atomic vapors was detected in Yu et al. (2019), confirming that the interaction is indeed long-range, and that it is present at any density.

According to the previous discussion, the interaction between atoms might be relevant to the determination of the quantum phase diagrams for the system constituted by $n$-level atoms interacting with $\ell$-modes of electromagnetic radiation in a cavity. The main objective of this work is to quantify the effect of the atomic dipole-dipole induced interaction on the properties of the ground state of the system. The original contributions of this work are the following: To establish the general dipole-dipole interaction Hamiltonian for a system of $n$-level atoms interacting with $\ell$-modes of electromagnetic radiation in a cavity. To calculate the associated energy surface, which allows us to determine the variational ground state, playing a fundamental role in finding the quantum phase diagrams of the system. The cases for $2$- and $3$- level atomic configurations are worked out explicitly, determining the quantum phase diagrams together with the corresponding order of the transitions. It is remarkable that, even for a finite number of atoms, the surface of maximum Bures distance is able to detect the phase transitions where the Ehrenfest method does not. Additionally we have found that the quantum phases continue to be dominated by a set of monochromatic regions as it was the case for noninteracting atoms, at least when the induced electric dipolar moments are real.

The paper is organized as follows: Section II derives the model for a system of $N_{a}$ identical $n$-level atoms interacting with $\ell$ modes of an electromagnetic field, including the atomic dipole-dipole interaction, and particularizes it for $2$- and $3$-level atoms. Section III constructs the variational energy surface from a complete set of test states which approach the quantum ground state, or any other quantum excited state. Here we focus on the ground state and study its phase diagram. This is applied in section IV to the case of $2$-level atoms, and in section V to the case of $3$-level atoms in their different atomic configurations, finding the critical values of the coupling parameters which minimize the energy, and determining the phase diagram in both attractive and repulsive scenarios of the atomic dipole-dipole interaction. In cases where a phase transition exists which defies the Ehrenfest classification, new criteria are proposed, one based on the second Casimir operator and another one based on the maximum Bures distance between neighboring states. Finally, section VI summarizes some conclusions. Two appendices present the matter collective operators and the atomic dipole-dipole operator in explicit form.

We consider a system of $N_{a}$ identical $n$-level atoms interacting with $\ell$ modes of a radiation field, placed in a cavity. The Hamiltonian is composed of three terms

where ${\bm{H}}_{D}$ is the diagonal contribution given by ($\hbar=1$)

the dipolar matter-field interaction is of the form Cordero et al. (2013)

In these expressions, $\Omega_{s}$ and ${\bm{\nu}}_{s}$ are the field frequency and photon number operator, respectively, of mode $s$; $\omega_{k}$ denotes the energy of the $k$-th atomic level with the convention $\omega_{j}<\omega_{k}$ for $j<k$); ${\bm{a}}_{s}^{\dagger}$ and ${\bm{a}}_{s}$ are the field creation and annihilation operators; and ${\bm{A}}_{jk}$ is the atomic transition operator between levels $k$ and $j$, which in a bosonic representation ${\bm{A}}_{jk}={\bm{b}}_{j}^{\dagger}{\bm{b}}_{k}$ plays the role of the collective matter operator obeying the unitary algebra in $n$ dimensions, $U(n)$; here, ${\bm{b}}^{\dagger}_{j}$ creates an atom in level $j$ and ${\bm{b}}_{k}$ annihilates one in level $k$ (cf. Eq. (54). The dipolar matter-field coupling intensity between field mode $s$ and atomic dipole formed by levels $j$ and $k$ is denoted by $\mu_{jk}^{(s)}$.

Atoms do not have permanent dipole moments in their ground state, as the center of charge of the electronic cloud coincides with that of the nucleus. In the presence of an electromagnetic field, however, these centers are displaced and the induced transition dipole moments are responsible for an atomic dipole-dipole interaction. In second quantization this dipole-dipole interaction takes the form

where bosonic creation ${\bm{b}}_{j}^{\dagger}$ and annihilation ${\bm{b}}_{k}$ operators were used. The factor $1/2$ in eq. (4) compensates the double accounting in the summation, as the particles are indistinguishable. The factor $(N_{a}-1)$ is included to have an interaction linear in the number of particles. The first index in the bra and the ket states corresponds to the first particle, while the second index corresponds to the second particle.

The set of operators that appear in (4) may be rewritten in terms of the collective matter operators, by means the bosonic commutation relation, as

where we have defined the oslash product between collective matter operators, which removes the self-interaction terms (see appendix A for more details).

The dipole-dipole interaction ${\bm{H}}_{dd}$ is obtained from the classical expression Jackson (1999) through the standard quantization procedure, which has the form

where $\vec{{\bm{d}}}_{i}=e\,\vec{{\bm{r}}}_{i},\ (i=a,b)$ are the induced vector operators of the electric dipole moments, $R$ is the separation between the dipoles and $\hat{n}=\vec{R}/R$ (with $\vec{R}=\vec{x}_{b}-\vec{x}_{a}$) the unitary vector in the direction from one dipole to another, at positions $\vec{x}_{a}$ and $\vec{x}_{b}$ (see Fig. 1). $\epsilon_{0}$ is the permittivity of vacuum, and for induced magnetic moments $\mu_{1},\,\mu_{2}$, the expression is the same with the replacements $d_{i}\to\mu_{i}$, and $\epsilon_{0}\to\mu_{0}$, the magnetic permeability of vacuum (cf., e.g., Cohen-Tannoudji et al. (heim)). Without loss of generality, we will consider here that the induced electric dipoles are real.

Thus, the two-body matrix elements in Hilbert space are

The indices of the dipole-dipole coefficient $g_{jj^{\prime}kk^{\prime}}$ refer to the two dipoles involved in the bra-ket (7).

For indistinguishable particles, and identifying the expansion components $\vec{d}_{jk}=\langle j|\vec{{\bm{d}}}_{a,b}|k\rangle$ of the dipolar operator in terms of the collective matter operators, viz. the dipolar operator given by $\vec{{\bm{d}}}=\sum_{j\neq k}^{n}\vec{d}_{jk}{\bm{A}}_{jk}$, the matrix element in Eq. (7) reads

where $R$ stands for the average distance between pairs of atoms. The hermiticity of Eq.(4) follows from the relations

Also, for real dipolar vectors $\vec{d}_{jk}=\vec{d}_{kj}$, one has $g_{jklm}=g_{jkml}=g_{kjlm}$.

Finally, using the oslash operator introduced above, we may write the dipole-dipole interaction in a simplified form as

Inserting the different contributions into (10), one may write the atomic dipole-dipole term in the Hamiltonian as (see appendix B)

where the operator ${\bm{W}}_{jk}^{2-{\rm levels}}$ stands for the dipole-dipole contribution of the pair $\vec{d}_{jk}\rightleftharpoons\vec{d}_{jk}$; the operator ${\bm{W}}_{jlk}^{3-{\rm levels}}$ for that of the pair of dipoles $\vec{d}_{jl}\rightleftharpoons\vec{d}_{lk}$ (here the atomic level $\omega_{l}$ plays the role of an intermediate level, so a prohibited dipolar transition $\vec{d}_{jk}=0$ is possible via the permitted dipolar transitions $\vec{d}_{jl}\neq 0$ and $\vec{d}_{lk}\neq 0$); and the operator ${\bm{W}}_{jklm}^{4-{\rm levels}}$ corresponds to the contribution of isolated dipoles $\vec{d}_{jk}\rightleftharpoons\vec{d}_{lm}$ (which do not share an energy level). The upper index denotes the number of different atomic levels which contribute to the interaction; hence, the terms ${\bm{W}}_{jlk}^{3-{\rm levels}}$ and ${\bm{W}}_{jklm}^{4-{\rm levels}}$ are zero for $n$-level atoms with $n=2$ and $n\leq 3$, respectively. The set of transitions included in each interaction term is given in table 1, and shown schematically in figure 2. These terms are given in appendix B. Also, the factors $1/2!$ and $1/4!$ in expression (11) eliminate the double summation due to index reordering.

Amongst the parameters in the Hamiltonian, we are free to choose $\omega_{1}=0$ and $\omega_{n}=1$, i.e., the energies are normalized to the highest atomic level. We also consider systems where only one field mode promotes the transition between a given pair of atomic levels; this constriction is imposed by the condition Cordero et al. (2016)

Since the interaction (11) involves the dipole-dipole contribution $g_{jklm}$, we have $g_{jklm}\neq 0$ only when $\mu_{jk}^{(s)}\neq 0$ and $\mu_{lm}^{(s^{\prime})}\neq 0$ for at least one of the modes $\Omega_{s}$ and $\Omega_{s^{\prime}}$. The first and second order Casimir operators [Eqs. (52 and 53)], whose eigenvalues are functions of the number of atoms $N_{a}$ and the number of levels $n$, are of course constants of motion.

As an example, we now write explicitly the contribution of ${\bm{H}}_{dd}$ for two- and three-level atoms:

The variational solution involves a test state which approaches the quantum ground or desired excited state, and which depends on a set of parameters $z_{i}$. The corresponding energy surface is obtained by taking the expectation value of the Hamiltonian and minimizing with respect to the parameters $z_{i}$ of the test state. In this work we focus on the ground state and take as test state the direct product of coherent states for both the matter and the field contributions. Clearly, this test state presents no matter-field entanglement, but it yields a good description of the minimum energy surface, as well as some expectation values of the physical quantities, and the phase diagram together with the order of the phase transitions.

For two-level atoms the expression of the energy surface reads

with $j<k$. The critical values of the corresponding energy surface Eq. (27) must satisfy

and, fixing $\phi_{j}=0$ and $\varrho_{j}=1$, one finds two solutions

Here, we have used a dimensionless matter-field coupling intensity $x_{jk}$, defined as

with $\mu_{jk}^{c}$ the critical value of the coupling constant when the atomic dipole-dipole interaction is neglected, and where $\omega_{jk}=|\omega_{k}-\omega_{j}|$; we have also defined

to simplify the notation.

For values $x_{jk}^{2}-y_{jk}\leq 0$ one finds only one critical value $\varrho_{k}=0$, for which the energy surface has the constant value $\omega_{j}$. When $x_{jk}^{2}-y_{jk}>0$ we have two critical values (32), in this case the energy surface has a dependence on the matter-field dipolar strength $x_{jk}$. After minimizing one finds

This relationship, $E_{\rm min}$ vs. $x_{jk}$, is shown in figure 3. The solid line (black) corresponds to the case $g=0$ without atomic dipole-dipole interaction; repulsive $g=0.05$ (dashed line, blue) and attractive $g=-0.05$ (dotted line, green) cases are also shown. Due to the minuteness of this interaction when compared with the dipolar matter-field interaction, the difference for dissimilar values of $g$ is difficult to appreciate. We have zoomed around the value $x_{jk}=1$ (see figure inset) where the transition into the collective region appears, in order to make this difference clear. In what follows, we will consider unnaturally large values for the atomic dipole-dipole coupling parameter $g$ so that its effect may be appreciated; when studying actual realistic systems these values (and their effects) must be scaled down accordingly.

The minimum energy for different (larger) values of the dipolar coupling strength is plotted in figure 4. For values of $g$ such that $y_{jk}>0$, the critical points $x_{jk}^{c}=\pm\sqrt{y_{jk}}$ divide the normal region $x_{jk}^{2}<(x_{jk}^{c})^{2}$ from the collective region $x_{jk}^{2}>(x_{jk}^{c})^{2}$. One should note that for the case without dipole-dipole interaction, $g=0$, (solid line in figure 4(a)) the critical points occur at $(x_{jk}^{c})^{2}=1$, while in the attractive case $g<0$ (dashed line in figure 4(a)) one has $(x_{jk}^{c})^{2}<1$, i.e., the normal region decreases. Correspondingly, for the repulsive case $g>0$ (dot-dash line in figure 4(a)) the normal region increases, as we have $(x_{jk}^{c})^{2}>1$. The anomalous behaviour is the strong attractive regime, this is characterised by values of $g$ such that $y_{jk}\leq 0$, when the normal region vanishes completely (dotted line in Fig. 4(a)). It is important to note that, for large matter-field coupling $x_{jk}^{2}\gg y_{jk}$, the minimum energy surface $E_{\rm min}$ tends to that without the atomic dipole-dipole interaction; in other words, the effect of the dipole-dipole terms on the energy surface is seen mainly in a vicinity of the normal region.

The order of the transition may be determined using the Ehrenfest classification Gilmore (1993), which involves the derivatives of the energy surface. We exemplify the case $g=0.5$ in figure 4(b), showing, respectively, the first (dashed-line) and second derivatives (solid line) of the energy. Since the second derivative presents a discontinuity at the critical point $x_{jk}^{c}$, a second order transition occurs at that location.

For three-level atomic systems interacting dipolarly with a two-mode electromagnetic field in a cavity, the atomic dipole-dipole interaction can be obtained from expression (10) or (59). For the case of real induced dipole moments one has only to consider the real coupling strengths $g_{1212},\,g_{1313},\,g_{2323}$ for two-level interactions, and $g_{1213},\,g_{1232},\,g_{1323}$ for those associated to three-level interactions. Thus the induced dipole-dipole interaction for three-level atoms takes the form,

Notice that for the different atomic configurations one has at most three real parameters; in the case of the $\Lambda$ configuration, for instance, we have the coupling strengths $g_{1313},\,g_{2323}$ and $g_{1323}$.

The corresponding variational energy surface for the dipole-dipole interaction may be obtained by taking the expectation value of (V) with respect the variational state $|\gamma_{1},\,\gamma_{2},\,\gamma_{3}\rangle\otimes|\alpha_{1},\,\alpha_{2}\rangle$, or from the general expression (28) by considering real induced dipole moments together with three-level atomic systems and a two-mode electromagnetic field. The resulting expressions for the $\Lambda$, $V$ and $\Xi$ atomic configuration are given by

For systems of $3$-level atoms interacting with two modes of electromagnetic field, the critical values of the phases (vide supra) are $\theta_{s}^{c}=0,\,\pi$ and $\phi_{k}^{c}=0,\,\pi$, for which the relationship $\mu_{jk}^{(s)}\cos(\theta_{s}^{c})\cos(\phi_{jk}^{c})>0$ is satisfied, and where we defined $\phi_{jk}^{c}=\phi_{k}^{c}-\phi_{j}^{c}$. Also, the critical values $r_{s}^{c}$ associated to the field are given as functions of the critical values $\varrho_{k}^{c}$ of the matter [cf. Eq. (29)]. These values must be calculated numerically, except when the dipole-dipole interaction is neglected, since in this latter case we have an analytical solution Cordero et al. (2015).

In this work we calculate the critical values for the three atomic configurations ($\Xi$, $\Lambda$ and $V$) and obtain the corresponding separatrix; we fix in all cases the double resonant condition, i.e., the field frequencies are given by $\Omega_{1}=\omega_{jk}$ and $\Omega_{2}=\omega_{lm}$. The atomic levels satisfy the condition $\omega_{1}<\omega_{2}<\omega_{3}$ with $\omega_{1}=0$ and $\omega_{3}=1$. We take $(j,k,l,m)=(1,2,2,3)$ and the value $\omega_{2}=3/4$ for the $\Xi$-configuration, $(j,k,l,m)=(1,3,2,3)$ and $\omega_{2}=1/4$ for the $\Lambda$-configuration, and $(j,k,l,m)=(1,2,1,3)$ and $\omega_{2}=3/4$ for the $V$-configuration. The values considered for the dipolar-dipolar strength $g_{jklm}$, assuming real dipolar vectors $\vec{d}_{jk}=\vec{d}_{kj}$, are given in table 2.

In order to exemplify how to obtain the separatrix, we consider explicitly the particular case of the $V$-configuration with a repulsive dipole-dipole strength $g_{3}$. The set of critical points $\varrho_{2}^{c}$ and $\varrho_{3}^{c}$ are evaluated numerically and inserted into the expression for the minimum energy; the result is shown in Fig. 5(a). The normal region, where $E_{min}=0$, is colored in black. The separatrix is found by calculating the first derivatives of the energy surface, as

which is shown in Fig. 5(b). It is a continuous surface. We calculate the second order derivative as

which is discontinuous [cf. Fig. 5(c)]. The loci form a separatrix which splits the normal from the collective region; in fact, this discontinuity shows that a second order transition occurs at these points for the $V$-configuration.

In Fig. 5(c), the slight undulation (observed by a small change in the orange hue of the surface) within the collective region in the second derivative of the minimum energy surface, is a signature of a kind of transition due to a change of subspaces formed by $2$-level atoms, as was discussed recently for the case without dipole-dipole interaction $g=0$ Cordero et al. (2015), from one subspace in which one of the radiation modes dominates to another subspace where the other mode dominates. This change grows as $g\to 0$, and gives a discontinuity when $g=0$. However, for values $g\neq 0$ the second derivative remains continuous, as well as derivatives of higher order; in other words, the Ehrenfest classification does not provide a criterion to determine that the transition exists. In this work, we propose to consider the second order Casimir operator corresponding to each $2$-level subsystem in order to label this transition (vide infra).

The second order Casimir operator for a system of $N_{a}$ particles of $n$-levels is given by

In particular, when only two levels are considered, we may define

which coincides with the second order Casimir operator for $2$-levels. Therefore, the expectation value $\langle\psi|C_{jk}|\psi\rangle$ will be close to $N_{a}(N_{a}+1)$ when the bulk of the contribution to the state $|\psi\rangle$ is given by the basis of the sub-system of the two levels $(j,k)$. Since the variational solution is independent of $N_{a}$, we fix for this calculation $N_{a}=2$ and consider the absolute value of the difference of the second order Casimir operator of each subsystem