A charged Coulomb Bose gas with dipole-dipole interactions

TL;DR

This paper investigates how dipole-dipole interactions influence the properties of a charged Coulomb Bose gas, revealing modifications in stability, thermodynamics, and coherence at weak coupling and finite temperatures.

Contribution

It introduces a systematic analysis of a charged Coulomb Bose gas with dipole-dipole interactions using the Hartree-Fock-Bogoliubov approach, highlighting new effects on system properties.

Findings

Dipole interactions affect collective excitations.

Condensate fraction and depletion are modified.

System stability and thermodynamics are influenced.

Abstract

We systematically study the properties of a charged Coulomb Bose gas with dipole-dipole interactions in the weak coupling limit at both zero and finite temperatures using the Hartree-Fock-Bogoliubov approach. We numerically analyze the collective excitations, the condensate fraction, the depletion, the chemical potential, and the static structure factor. Moreover, we compare our new findings with those of nondipolar charged Coulomb Bose gas. Our results reveal that the complex interplay of Coulomb and dipole-dipole interactions may modify the stability, the thermodynamics and the coherence of the system.

Equations

$$\displaystyle i\hbar\dot{\Phi}({\bf r},t)=\bigg{(}-\frac{\hbar^{2}\nabla^{2}}{2m}-\mu\bigg{)}\Phi({\bf r},t)+\int d{\bf r^{\prime}}V({\bf r}-{\bf r^{\prime}})\bigg{[}n({\bf r^{\prime}},t)\Phi({\bf r},t)+\tilde{n}({\bf r},{\bf r^{\prime}},t)\Phi({\bf r^{\prime}},t)+\tilde{m}({\bf r},{\bf r^{\prime}},t)\Phi^{*}({\bf r^{\prime}},t)\bigg{]},$$

$$V(\mathbf{r}-{\mathbf{r}}^{\prime })$$

$$=g\delta (\mathbf{r}-{\mathbf{r}}^{\prime })+\frac{C_{dd}}{4\pi }\frac{1-3{\cos }^2\theta }{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}+\frac{C_c}{|\mathbf{r}-{\mathbf{r}}^{\prime }|},$$

$$\tilde{V}(\mathbf{k})$$

$$\displaystyle=g\big{[}1+\epsilon_{\text{dd}}(3\cos^{2}\theta_{k}-1)+\epsilon_{c}/k^{2}\big{]},$$

$$\displaystyle\mu=\tilde{V}(|\mathbf{k}|=0)n+\frac{1}{V}\sum\limits_{\bf k\neq 0}\tilde{V}(\mathbf{k})\big{(}\tilde{n}_{k}+\tilde{m}_{k}\big{)},$$

$${\tilde{n}}_k=[v_k^2+(u_k^2+v_k^2)N_k],$$

$${\tilde{m}}_k=u_k{v}_k(2N_k+1),$$

$$u_k^2=\frac{{\omega }_k+{\epsilon }_k}{2{\epsilon }_k},v_k^2=\frac{{\omega }_k-{\epsilon }_k}{2{\epsilon }_k},$$

$${\epsilon }_k=\sqrt{{\omega }_k^2-{\Delta }_k^2},$$

$${\omega }_k\equiv E_k+n\tilde{V}(|\mathbf{k}|=0)+n_c\tilde{V}(\mathbf{k})+\frac{1}{V}\sum _{p\ne 0}{\tilde{n}}_p\tilde{V}(\mathbf{k}+\mathbf{p})-{\mu }_1,$$

$${\Delta }_k\equiv n_c\tilde{V}(\mathbf{k})+\frac{1}{V}\sum _{p\ne 0}{\tilde{m}}_p\tilde{V}(\mathbf{k}+\mathbf{p}),$$

$${\mu }_1=n\tilde{V}(|\mathbf{k}|=0)+\frac{1}{V}\sum _{k\ne 0}\tilde{V}(\mathbf{k})({\tilde{n}}_k-{\tilde{m}}_k).$$

$${\epsilon }_q={\epsilon }_g\sqrt{\frac{q^4}{12r_s}+\frac{q^2}{3r_s}\frac{r_0^2}{{\xi }^2}(1-{ϵ}_{dd})+1},$$

$$\tilde{n}=\frac{1}{2}\int \frac{d\mathbf{k}}{(2\pi {)}^3}\left[\frac{E_k+n\tilde{V}(\mathbf{k})}{{\epsilon }_k}\text{coth}\left({\epsilon }_k/2T\right)-1\right],$$

$$\tilde{m}=-\frac{1}{2}\int \frac{d\mathbf{k}}{(2\pi {)}^3}\frac{n\tilde{V}(\mathbf{k})}{{\epsilon }_k}\text{coth}\left({\epsilon }_k/2T\right).$$

$${\mu }_{\text{LHY}}=\frac{1}{2}\int \frac{d\mathbf{k}}{(2\pi {)}^3}\tilde{V}(\mathbf{k})\left[\frac{E_k}{{\epsilon }_k}\text{coth}\left({\epsilon }_k/2T\right)-1\right].$$

$$S(\mathbf{k})=\frac{E_k}{{\epsilon }_k}\coth \left(\frac{{\epsilon }_k}{2T}\right),$$

Full text

The last decades have witnessed a remarkable surge of interest in charged bosons. A charged Bose gas (CBG) in which particles interact via Coulomb forces was stimulated by potential applications in statistical mechanics Wu , physics of high-temperature superconductivity Alex , Meissner-Ochsenfeld effect Shf1 ; Shf2 , collective excitations Bohm ; Bon , nuclear reactions in dense plasmas and astrophysics Nin ; Ginz ; Hans ; Schr ; Mul , Wigner crystallization Wig ; Cer ; Pel ; Drum and so on.

The properties of CBG at zero temperature have been extensively studied using different approaches. In 1961, Foldy Foldy calculated the ground-state energy and the elementary excitations spectrum of CBG employing the Bogoliubov theory Bog , valid only at very low temperatures and in the high-density (weak coupling) regime, $r_{s}\ll 1$ Foldy . The coupling strength is characterized by the dimensionless gas parameter $r_{s}=r_{0}/a_{B}$, where $a_{B}$ is the Bohr radius, and $r_{0}=(3/4\pi n)^{1/3}$ is the interparticle separation with $n$ being the mean density. Higher-order corrections to the ground-state energy were obtained in Lee ; Bru ; Woo using beyond the Bogoliubov approximation. The subsequent studies dealt with the random phase approximation dielectric response function Sing ; Vash ; Hir , quantum-to-classical mappings Perr ; Sand , collective modes and screening properties of CBG Alex1 ; Alex2 . Further investigations have been performed at finite temperature focusing on the critical temperature, elementary excitations, the normal and anomalous momentum distributions Hans ; Fett ; Bish ; Hore ; Strep ; Dav ; Dav1 . Rigorous results for various ground-state properties of CBG have been obtained in the frame of Quantum Monte Carlo methods (see e.g. Cer ; Mor ; Nor ; Palo and references therein). Ground-state energies of the two-component CBG have been computed by Lieb et al.Lieb using Dyson’s method. The authors of Refs.Gu1 ; Gu2 have analyzed the magnetic properties of CBG within the mean-field theory. Other aspects of CBG have been studied in Luk and references therein.

In this Letter we investigate the ground-state properties of CBG with short-range (contact) and dipole-dipole interactions (DDI) using the full HFB theory. Meissner effect in a charged Bose gas with short-range repulsion has been addressed in Shun , where the collective excitations due to the repulsive interaction found to complicate the situation. Ultracold quantum gases with DDI have attracted tremendous interests recently due to the long-range character and the anisotropy (see for review Pfau ; Carr ; Baranov ; Pupillo2012 and references therein) in contrast to the short-range interactions. Dipolar Bose-Einstein condensates (BECs) consist of atoms with sizeable magnetic dipole moments and have been experimentally realized with 52Cr Gries , 164Dy ming , and 168Er erbium . Most recently, Er-Dy mixture has been experimentally achieved in two-species magneto-optical trap Ilz . The DDI may strongly affect the excitations, the dynamics and the thermodynamic properties of the BEC Pfau ; Carr ; Baranov ; Pupillo2012 . In addition, they lead to the emergence of novel quantum phases such as supersolid and droplet states (see for review Luo ; Pfau2 ; Guo and references therein). Therefore, it is instructive to discuss the role played by the competition between the Coulomb, contact and dipolar interactions in the CBG.

Within the Hartree-Fock-Bogoliubov (HFB) theory we write down the generalized nonlocal Gross-Piteavskii and calculate the Bogoliubov excitations energy. We show that this latter presents a plasmon gap spectrum at long wavelengths (low momenta) regime due to the Coulomb interactions. The presence of DDI lead to shift the frequency of such a gap. We provide useful analytic expressions for the normal and anomalous fluctuations, the equation of state (EoS), and the static structure factor. The obtained expressions (notably those of the anomalous density and the EoS) suffer from both infrared and ultraviolet divergences. The former originates from the Coulomb interaction while the latter is caused by the use of a contact interaction. In the absence of contact interactions and DDI, exact cancellation of infrared-divergent terms in the HFB shift of the single-particle excitation energy has been demonstrated in Ref.Dav1 .

Numerical results of the obtained equations are presented in terms of temperature, the DDI, and the gas parameter in the weak-coupling regime. We show that the normal and anomalous fractions increase with $r_{s}$. In a comparison with a conventional dipolar BEC this study reveals that the DDI can decrease both the noncondensed and the anomalous concentrations apart in the regime of very weak coupling where the depletion rises with the DDI. Our results indicate also that the condensed fraction reduces with $r_{s}$ for any value of temperature. Crucially, we point out that the correction to the EoS arising from the quantum fluctuations exhibits an unconventional behavior with temperature, DDI and the gas parameter. Furthermore, it is shown that the static structure factor overshoots unity displaying a sharp peak at lower temperatures and for relatively large DDI. It is found that the interplay of the Coulomb interaction and the DDI may lead to shift the height and the position of such peaks. To the best of our knowledge, this is the first work unveiling these spectacular properties of CBG.

We consider a gas of $N$ identical charged bosons with charge $e$ and mass $m$ in a box of volume $V$ with both contact and dipolar interactions moving in a static uniform neutralizing background. We assume that the dipoles are strictly aligned along the $z$-axis, in this case the interaction potential has a contact component related to the $s$-wave scattering length $a$ and a dipolar component (see below). In the frame of the HFB theory, uniform charged bosons with DDI are described by the following nonlocal generalized Gross-Piteavskii equation Dav ; Boudj4 ; Boudj2 ; Boudj3 :

[TABLE]

where $\Phi({\bf r})=\langle\hat{\psi}({\bf r})\rangle$ is the condensate wavefunction, with $\hat{\psi}({\bf r})$ being the boson field operator, $\mu$ is the chemical potential, $n_{c}({\bf r})=|\Phi({\bf r})|^{2}$, $\tilde{n}({\bf r})=\langle\hat{\bar{\psi}}^{\dagger}({\bf r})\hat{\bar{\psi}}({\bf r})\rangle$ and $\tilde{m}({\bf r})=\langle\hat{\bar{\psi}}({\bf r})\hat{\bar{\psi}}({\bf r})\rangle$ are respectively the condensed, noncondensed and anomalous densities, where $\hat{\bar{\psi}}({\bf r})=\hat{\psi}({\bf r})-\Phi({\bf r})$ is the noncondensed part of the field operator. The total density is given by $n({\bf r})=n_{c}({\bf r})+\tilde{n}({\bf r})$. The terms $\tilde{n}({\bf r,r^{\prime}})$ and $\tilde{m}({\bf r,r^{\prime}})$ are respectively, the normal and the anomalous one-body density matrices which account for the exchange interaction between the condensed and noncondensed atoms. The two-body interaction potential is

[TABLE]

where $g=4\pi\hbar^{2}a/m>0$ is the coupling constant corresponds to the contact interaction with $a$ being the $s$-wave scattering length, $C_{dd}=d^{2}/\epsilon_{0}$ is the electric DDI strength with $\epsilon_{0}$ being the permittivity of vacuum, $\theta$ is the angle between the polarization direction and $\bf r$, $C_{c}$ describes the strength of Coulomb interactions, it is related to the Bohr radius via: $C_{c}=4\pi\hbar^{2}/(ma_{B})$ Foldy ; Dav1 ; Alex1 . It is clear that for $C_{c}=C_{\text{dd}}=0$, Eq.(1) reduces to the standard local Gross-Piteavskii equation.

Now we calculate the elementary excitations and fluctuations of a CBG. In the high-density limit and when the temperature is close to zero, we can linearize Eq.(1) using the $\Phi=\sqrt{n_{c}}+\delta\Phi$, where $\delta\Phi({\bf r},t)=u_{k}e^{i{\bf k\cdot r}-i\varepsilon_{k}t/\hbar}+v_{k}e^{i{\bf k\cdot r}+i\varepsilon_{k}t/\hbar}\ll\sqrt{n_{c}}$, where $u_{k}$ and $v_{k}$ are the Bogoliubov amplitudes. In Fourier space, the wavefunctions are real-valued ($\Phi_{0}=\Phi_{0}^{*}=\sqrt{n_{c}}$), and the interaction potential (2) is written as:

[TABLE]

where $\epsilon_{\text{dd}}=C_{\text{dd}}/3g$ is the relative strength which describes the interplay of contact interaction and the DDI, $\theta_{k}$ is the angle between the vector $\mathbf{k}$ and the polarization direction, and $\epsilon_{c}=C_{c}/g$ which has dimension (length)-2 is the relative coupling strength which describes the interplay of contact interaction and Coulomb interaction. For the most widely utilized species of cold atoms (such as Rb, Cr, Er, Dy), $\epsilon_{c}a_{B}^{2}$ ranges from 0.005 to 0.01. Due to the electroneutrality, one can set $\tilde{V}_{c}(\mathbf{k}\equiv 0)=0$, which is a consequence of the compensation of the boson-boson repulsion by the attraction due to a spatially homogeneous charged background Alex2 .

The chemical potential is given according to Eq.(1) by Boudj5 ; Yuk

[TABLE]

where $\tilde{n}_{k}$ and $\tilde{m}_{k}$ stand for the normal and anomalous distributions which can be defined in the spirit of the HFB approximation as Dav ; Yuk ; Boudj5 :

[TABLE]

and

[TABLE]

where $N_{k}=\langle\hat{b}_{k}^{\dagger}\hat{b}_{k}\rangle=[\exp(\varepsilon_{k}/T)-1]^{-1}$ are occupation numbers for the excitations. The solution of the resulting Bogoliubov-de-Gennes (BdG) equations gives for the Bogoliubov quasiparticle amplitudes

[TABLE]

and for the Bogoliubov excitations energy Boudj5 ; Yuk

[TABLE]

where

[TABLE]

and

[TABLE]

where $E_{k}=\hbar^{2}k^{2}/2m$ is the free particle energy.

Evidently, the HFB spectrum (7) has an unphysical gap in the limit of long wavelengths due to the inclusion of the anomalous correlations. To circumvent this problem, we define the chemical potential $\mu_{1}$ as Griffin ; Yuk ; Boudj :

[TABLE]

with the condition of charge neutrality which cancels the $k=0$ term associated with Coulomb interaction in the sum Dav ; Alex2 . It is obvious that the chemical potential $\mu_{1}$ of Eq.(8) renders the spectrum (7) gapless in agreement with the Hugenholtz-Pines theorem HP . Importantly, the excitation spectrum (7) has a roton-maxon structure originating from the anisotropy of the Hartree-Fock corrections.

Setting $g=C_{\text{dd}}=0$ in Eq.(7), the HFB excitation energy for a charged Bose gas is well reproduced Dav . In the long-wavelength limit $k\rightarrow 0$ where $\lim\limits_{k\rightarrow 0}(\tilde{n}_{k}-\tilde{m}_{k})\simeq-1/2$ Dav , the zero-temperature Bogoliubov excitations energy (7) coincides with the plasma energy $\varepsilon_{g}/\hbar=\sqrt{nC_{c}/m}$ (i.e. plasmon gap). At finite temperatures, one can expect that the spectrum energy (7) increases with $T$ and vanishes at the transition. In the high momenta limit, $k\rightarrow\infty$, the excitations spectrum (7) reduces to the free particle law ($\varepsilon_{k}=E_{k}$).

For the sake of simplicity, we shall assume from now on that $\tilde{m}/n_{c}\ll 1$ and $\tilde{n}/n_{c}\ll 1$ (i.e. we neglect higher-order terms) and set $\theta_{k}=\pi/2$. In such a case the Bogoliubov excitations energy (7) reduces to the following dimensionless form:

[TABLE]

where $q=kr_{0}$, and $\xi=\hbar/\sqrt{mng}$ is the standard healing length of the condensate.

In Figure 1 we show the behavior of the Bogoliubov spectrum for different values of the coupling parameter $r_{s}$. It is clearly seen that the spectrum $\varepsilon_{q}$ increases monotonically with $q$ whatever the values of $r_{s}$ and $\epsilon_{dd}$. It increases with decreasing both $r_{s}$ and $\epsilon_{dd}$.

In a homogeneous Bose gas, the depletion and the anomalous density are defined as Dav ; Boudj5 ; Yuk : $\tilde{n}=V^{-1}\sum\limits_{\bf k\neq 0}n_{k}$, and $\tilde{m}=-V^{-1}\sum\limits_{\bf k\neq 0}\tilde{m}_{k}$. Working in the thermodynamic limit, the sum over $k$ can be replaced by integrals according to the prescription $\sum\limits_{\bf k}\rightarrow V\int_{0}^{\infty}d{\bf k}/(2\pi)^{3}$. Thereafter inserting the identity $2N(x)+1=\coth(x/2)$ into Eqs.(5) and (6), we then obtain for the noncondensed and anomalous densities Boudj1 ; Boudj5 :

[TABLE]

and

[TABLE]

The validity of the present HFB approach requires the inequality: $\tilde{n}\ll n$. This implies that to have a dilute CBG, the following conditions: $r_{s}\ll 1$ and $na^{3}\ll 1$ must be fulfilled.

The condensed fraction can be evaluated through: $n_{c}/n=1-\tilde{n}/n$. For $g=C_{dd}=0$, the condensed fraction reduces to $n_{c}/n\simeq 1-0.2\,r_{s}^{3/4}$ Foldy .

Correction to the chemical potential due to the Lee-Huang-Yang (LHY) quantum fluctuations can be given from Eq.(4) as Boudj1 ; Boudj5 : \mu_{\text{LHY}}=V^{-1}\sum\limits_{\bf k\neq 0}\tilde{V}(\mathbf{k})\big{(}\tilde{n}_{k}+\tilde{m}_{k}\big{)}. Then using the definitions $\tilde{n}_{k}$ and $\tilde{m}_{k}$ from Eqs.(5) and (6), we obtain:

[TABLE]

This equation permits us to calculate the LHY corrections to all thermodynamic quantities. In the absence of the short-range and dipolar interactions the LHY-corrected EoS reads $\mu_{\text{LHY}}\simeq-0.23\,\varepsilon_{g}r_{s}^{3/4}$ Foldy .

It is worth stressing that the exact analytical solutions of the integrals (10), (11) and (12) are not trivial except in some limiting cases. Therefore, we solve them numerically.

The results for the noncondensed and the anomalous fractions for different values of $r_{s}$ and $\epsilon_{dd}$ are shown graphically in Fig.2. We see that the noncondensed and the anomalous fractions increase with the coupling strength $r_{s}$ even in the absence of the DDI ($\epsilon_{dd}=0$). In the very weak coupling regime $r_{s}\lesssim 0.15$, $\tilde{n}/n$ rises with $\epsilon_{dd}$ (see the inset of Fig.2.a). On the contrary, for $r_{s}>0.15$, the depletion decreases with the DDI even for relatively large $\epsilon_{dd}$. This can be attributed to the effect of the Coulomb interaction which dominates both the short-range and the dipolar interactions. The anomalous fraction is decreasing with the DDI in the whole range of $r_{s}$ as is seen in Fig.2.b. For very small $r_{s}$, one has $\tilde{n}/n\ll 1$ and $\tilde{m}/n\ll 1$ in good agreement with the results of the Bogoliubov Dav and with Monte Carlo data Mor . Whereas for larger $r_{s}$, both the depletion and the anomalous fraction increase continuously whatever the value of $\epsilon_{dd}$. For example, for $r_{s}=0.5$ and $\epsilon_{dd}=0.45$, one has $\tilde{n}/n>10\%$ and $\tilde{m}/n\gtrsim 25\%$, pointing out that the present HFB theory becomes no longer applicable in the regime of $r_{s}\gtrsim 0.5$. Another important remark is that the anomalous fraction is larger than the normal fraction similarly to the neutral atomic dipolar and nondipolar BECs Yuk ; Boudj5 ; Boudj ; Griffin ; Boudj1 .

Figure 3.a reports the condensate fraction $n_{c}/n=1-\tilde{n}/n$ as a function of temperature $T/\varepsilon_{g}$ for different values of the coupling strength $r_{s}$ and of the relative interaction strength $\epsilon_{dd}$. It is clearly visible that the condensate fraction strongly decreases with $r_{s}$ for any values of temperature. This indicates that the dipolar CBG becomes strongly depleted for large $r_{s}$ and $\epsilon_{dd}$ (see red lines). We see also that at fixed temperature, $n_{c}/n$ lowers with increasing $\epsilon_{dd}$ except for large values of $r_{s}$, where it slightly decreases with decreasing $\epsilon_{dd}$ due to the interplay of the Coulomb and dipolar interactions.

Figure 3.b depicts that at low temperatures $T\lesssim\varepsilon_{g}$, the LHY-corrected chemical potential reduces with $r_{s}$ and $\epsilon_{dd}$. Remarkably, it remains negative in such a regime leading to decrease the total EoS. The negative value of $\mu_{\text{LHY}}$ is a product of oppositely charged background Alex1 . At $T\gtrsim\varepsilon_{g}$, $\mu_{\text{LHY}}/\varepsilon_{g}$ increases linearly with temperature especially for large $r_{s}$ regardless of the value of $\epsilon_{dd}$. This can be attributed to the competition between the contact, the dipole-dipole and Coulomb interactions.

Information on the coherence and on the fluctuations of CBG are contained in the static structure factor which is defined as the Fourier transform of the density-density correlation function $S({\bf k})=\langle\delta\hat{n}({\bf k})\delta\hat{n}(-{\bf k})\rangle/n$ PitaevString , where $\delta\hat{n}({\bf k})=\int d{\bf r}\,\delta\hat{n}({\bf r})e^{-i.\bf k.r}$ and \delta\hat{n}({\bf r})=\sqrt{n({\bf r})}\sum_{k}\big{\{}[u_{k}({\bf r})-v_{k}({\bf r})]\exp(-i\varepsilon_{k}t/\hbar)\,\hat{b}_{k}+H.c\big{\}}. In terms of the elementary excitation energy $S({\bf k})$ reads PitaevString :

[TABLE]

At zero temperature, Eq.(13) reduces to $S({\bf k})=E_{k}/\varepsilon_{k}$. At low temperatures and in the phonon regime ($k\rightarrow 0$) one has $S({\bf k})\simeq T/\varepsilon_{g}$. In the opposite situation, at higher $T$, $S({\bf k})$ simplifies to its zero temperature value except in the limit $k\rightarrow 0$ where the structure factor approaches its asymptotic value PitaevString . A non-correlated gas has a structureless spectrum $S(k)=1$.

The numerical solution of Eq.(13) is presented in Fig.4. As one can see from the figure, at low temperatures $T<\varepsilon_{g}$ where the main contribution comes from low momenta, the static structure factor has a strong dependence on the temperature, the Coulomb interaction and on the DDI. We observe also that S(k) increases significantly and develops a pronounced peak around $q=q_{0}$. For instance, for $r_{s}=0.5$, $q_{0}\simeq 1.6$ or equivalently $k\simeq 1.6/r_{0}$ (see Fig.4.a). The position of such a peak relies on $r_{s}$. Remarkably, S(k) develops a peak even at relatively high temperatures ($T\simeq\varepsilon_{g}$) for all $r_{s}$ due to the interplay of Coulomb and dipolar interactions. Augmenting further the temperature ($T\geq 2\varepsilon_{g}$), thermal effects do not favor such a localization behavior of the particles and the static structure factor is almost monotonic (see blue lines in Fig.4).

However in the case of nondipolar CBG ($\epsilon_{dd}=0$), the structure factor becomes less significant and exhibits a peak only at low temperatures and for $r_{s}>0.02$ in contrast to the dipolar CBG (see the Fig.4.b). This can be understood due to the fact that in the absence of DDI and for $r_{s}>0.02$, the Coulomb interaction dominates the system leading to strong thermal fluctuations even at $T\lesssim\varepsilon_{g}/2$, giving rise to destroy the coherence of the system.

In conclusion, we studied the ground-state properties of a weakly interacting CBG with DDI at both zero and finite temperatures using the self-consistent HFB theory. We derived the generalized nonlocal Gross-Pitaevskii equation that describe the dynamics and the equilibrium of such a system. By solving the BdG equations we analyzed the Bogoliubov excitations spectrum that exhibits a plasmon gap giving rise to infrared divergences arising from the Coulomb interaction. Furthermore, the normal and anomalous fractions, the EoS, and the static structure factor have been computed numerically. We show that the intriguing interplay of Coulomb interactions and the DDI leads to affect these quantities and thus plays a pivotal role in the physics of the system.

Even though the experimental realization of identical charged bosons with DDI is a challenging problem, they constitute a promising area for applications. In the limit of a strong (even intermediate) coupling, one can expect that the presence of the DDI may lead to the appearance of a very sharp peak in the static structure factor driving the system to a transition to a Wigner crystal phase. A qualitative study of this crystallization necessitates sophisticated tools such as Quantum Monte Carlo simulations. An important extension of our work would be the study of the Meissner-Ochsenfeld effect in dipolar CBG. The competition between repulsive short-range, Coulomb and dipolar interactions may enhance the coherence of the system implying a singularity in the susceptibility prior to the BEC phase Shun . This could be a signature of the emergence of such a Meissner-Ochsenfeld effect.