Dipolar Bose gas with three-body interactions at finite temperature

TL;DR

This paper studies how three-body interactions influence the properties of a trapped dipolar Bose gas at finite temperature, focusing on transition temperature, condensate fraction, and collective excitations.

Contribution

It introduces a detailed analysis of three-body contact interactions in dipolar Bose gases at finite temperature using the Hartree-Fock-Bogoliubov approximation, highlighting their effects on various physical properties.

Findings

Three-body interactions modify the transition temperature.

Anomalous pair correlations are affected by temperature.

Collective mode behaviors are influenced by three-body effects.

Abstract

We investigate effects of three-body contact interactions on a trapped dipolar Bose gas at finite temperature using the Hartree-Fock-Bogoliubov approximation. We analyze numerically the behavior of the transition temperature and the condensed fraction. Effects of the three-body interactions, anomalous pair correlations and temperature on the collective modes are discussed.

Equations

$$i\hslash \frac{\partial \Phi (\mathbf{r},t)}{\partial t}$$

$$+\left[g_2\tilde{m}(\mathbf{r},t)+\frac{3g_3}{2}\tilde{m}(\mathbf{r},t)n_c(\mathbf{r},t)\right]{\Phi }^{\ast }(\mathbf{r},t)$$

$$\displaystyle+\int d{\bf r^{\prime}}V_{d}({\bf r}-{\bf r^{\prime}})\bigg{[}n({\bf r^{\prime}},t)\Phi({\bf r},t)+\tilde{n}({\bf r},{\bf r^{\prime}},t)\Phi({\bf r^{\prime}},t)+\tilde{m}({\bf r},{\bf r^{\prime}},t)\phi^{*}({\bf r^{\prime}},t)\bigg{]},$$

$${\epsilon }_k{u}_k(\mathbf{r})$$

$$-{\epsilon }_k{v}_k(\mathbf{r})$$

$$\tilde{n}(\mathbf{r},{\mathbf{r}}^{\prime })$$

$$\displaystyle+v_{k}({\bf r^{\prime}})v_{k}^{*}({\bf r})\bigg{\}},$$

$$\tilde{m}(\mathbf{r},{\mathbf{r}}^{\prime })$$

$$\displaystyle+u_{k}({\bf r^{\prime}})v_{k}^{*}({\bf r})\bigg{\}},$$

$$g_2|\Phi {|}^2\Phi +g_2\tilde{m}{\Phi }^{\ast }+\frac{3g_3}{2}{n}_c\tilde{m}{\Phi }^{\ast }$$

$$\displaystyle=g_{2}\bigg{[}1+\frac{\tilde{m}(1+3g_{3}n_{c}/g_{2})}{\Phi^{2}}\bigg{]}|\Phi|^{2}\Phi$$

$$=g_R|\Phi {|}^2\Phi .$$

Full text

I Introduction

Recently, Bose-Einstein condensates (BEC) with dipole-dipole interactions (DDI) have received considerable attention both experimentally and theoretically Baranov ; Pfau ; Carr ; Pupillo2012 . Dipolar quantum gases, in stark contrast to dilute gases of isotropic interparticle interactions, offer fascinating prospects of exploring ultracold gases and novel many-body quantum phases with atomic interactions that are long-range and spatially anisotropic. Impressionnant efforts has been directed towards the ground states properties and elementary excitations of such dipolar systems at both zero and finite temperatures Santos ; Santos1 ; Dell ; Eberlein ; Bon1 ; Bon ; Corm ; He ; Biss ; lime ; Boudj ; Boudj1 ; Boudj2 .

Three-body interactions (TBI) are ubiquitous and play an important role in a wide variety of interesting physical phenomena, and yield a new physics and many surprises not encountered in systems dominated by the two-body interactions. Recently, many experimental and theoretical techniques have been proposed to observe and realize the TBI in ultracold Bose gas Ham ; Will ; Daly ; Petrov . For instance, inelastic three-body processes, including observations of Efimov quantum states and atom loss from recombination have been also reported in Refs Eff ; Eff1 ; Bed ; Kra ; Brut . Few-body forces induce also nonconventional many-body effects such as quantum Hall problems Grei and the transition from the weak to strong-pairing Abelian phase Moor ; Read . In 2002, Bulgac Bulg predicted that both weakly interacting Bose and Fermi gases with attractive two-body and large repulsive TBI may form droplets. Dasgupta Dasg showed that if the two-body interactions were attractive, the presence of the TBI leads to a nonreversible BCS-BEC crossover. Furthermore, many proposals dealing with effects of effective TBI in ultracold bosonic atoms in an optical lattice or a superlattice are reported in Daly1 ; Mazz ; Singh ; Mahm . Moreover, the TBI in dilute Bose gases may give rise to considerably modify the collective excitations at both zero and finite temperatures Abdul ; Hamid ; Chen , the transition temperature, the condensate depletion and the stability of a BEC in a one (1D)- and two (2D)-dimensional trapping geometry Peng ; Mash .

However, little attention has been paid to effects of TBI on dipolar BECs. For instance, it has been argued that TBI play a crucial role in the stablization of the supersolid state in 2D dipolar bosons Petrov1 and in the quantum droplet state in 3D BEC with strong DDI Pfau1 ; Kui ; Blakie ; BoudjDp .

Our main aim here is to study effects of the TBI on weakly interacting dipolar Bose gases in a pencake trap at finite temperature. To this end, we employ the full Hartree-Fock-Bogoliubov (HFB) approximation. This approach which takes into account the pair anomalous correlations has been extensively utilized to describe the properties of both homogenous and trapped BECs with contact interactions Hut . We will show in particular how the interplay between the DDI, TBI and temperature can enhance the density profiles, the condensed fraction and the collective modes of the system.

The rest of the paper is organized as follows. In Sec.II, we introduce the full HFB formalism for dipolar BECs with TBI. We discuss also the issues encountered in our model and present the resolution of these problems. Section III is devoted to presenting and discussing our numerical results. Our conclusions are drawn in Sec.IV.

II Three-body model for dipolar bosons

Consider a dipolar BEC with contact repulsive two-body interactions and TBI confined in a pancake-shaped trap with the dipole moments of the particles oriented perpendicular to the plane. It is straightforward to check that the condensate wavefunction $\Phi({\bf r})=\langle\hat{\psi}({\bf r})\rangle$, with $\hat{\psi}({\bf r})$ being the Bose field operator, satisfies the generalized Gross-Pitaevskii (GP) equation BoudjDp

[TABLE]

where $h^{sp}=-\hbar^{2}\Delta/2m+U({\bf r})$ is the single particle Hamiltonian, $m$ is the particle mass, $U({\bf r})=m\omega_{\rho}^{2}(\rho^{2}+\lambda^{2}z^{2})/2$, $\rho^{2}=x^{2}+y^{2}$, $\lambda=\omega_{z}/\omega_{\rho}$ is the ratio between the trapping frequencies in the axial and radial directions. The two-body coupling constant is defined by $g_{2}=4\pi\hbar^{2}a/m$ with $a$ being the $s$-wave scattering length which can be adjusted using a magnetic Feshbach resonance. The three-body coupling constant $g_{3}$ is in general a complex number with $Im(g_{3})$ describing the three-body recombination loss and $Re(g_{3})$ accounting for the three-body scattering parameter. In the present paper, we do not consider the three-body recombination terms i.e. $Im(g_{3})=0$, so the system is stable which is consistent with recent experiments Kra ; Pfau1 . The DDI potential is $V_{d}({\bf r})=C_{dd}(1-3\cos^{2}\theta)/(4\pi r^{3})$, where $C_{dd}={\cal M}_{0}{\cal M}^{2}(=d^{2}/\epsilon_{0})$ is the magnetic (electric) dipolar interaction strength, and $\theta$ is the angle between the relative position of the particles ${\bf r}$ and the direction of the dipole. The condensed and noncondensed densities are defined, respectively as $n_{c}({\bf r})=|\Phi({\bf r})|^{2}$, $\tilde{n}({\bf r})=\langle\hat{\bar{\psi}}^{\dagger}({\bf r})\hat{\bar{\psi}}({\bf r})\rangle$ and $n({\bf r})=n_{c}({\bf r})+\tilde{n}({\bf r})$ is the total density. The term $\tilde{n}({\bf r,r^{\prime}})$ and $\tilde{m}({\bf r,r^{\prime}})$ are respectively, the normal and the anomalous one-body density matrices which account for the dipole exchange interaction between the condensate and noncondensate.

Equation (1) describes the coupled dynamics of the condensed and noncondensed components. For $g_{3}=0$, it recovers the generalized nonlocal finite-temperature GP equation with two-body interactions. For $\tilde{m}=0$, Eq.(1) reduces to the HFB-Popov equation Bon ; Corm ; BoudjDp which is gapless theory. For $\tilde{m}=\tilde{n}=0$, it reduces to standard GP equation that describes dipolar Bose gases only at zero temperature.

Upon linearizing Eq.(1) around a static solution $\Phi_{0}$, utilizing the parameterization $\Phi({\bf r},t)=[\Phi_{0}({\bf r})+\delta\Phi({\bf r},t)]e^{-i\mu t/\hbar}$, where $\delta\Phi=\sum_{k}[u_{k}({\bf r})e^{-i\varepsilon_{k}t/\hbar}+v_{k}({\bf r})e^{i\varepsilon_{k}t/\hbar}]$, and $\varepsilon_{k}$ is the Bogoliubov excitations energy. The quasi-particle amplitudes $u_{k}({\bf r}),v_{k}({\bf r})$ satisfy the generalized nonlocal Bogoliubov-de-Gennes (BdG) equations BoudjDp :

[TABLE]

where $\hat{\cal L}=h^{sp}+2g_{2}n({\bf r})+3g_{3}[n_{c}^{2}({\bf r})+4n_{c}({\bf r})\tilde{n}({\bf r})+\tilde{m}^{*}({\bf r})\Phi^{2}({\bf r})+\tilde{m}({\bf r})\Phi^{*2}({\bf r})]/2+\int d{\bf r^{\prime}}V_{d}({\bf r}-{\bf r^{\prime}})n({\bf r^{\prime}})-\mu$, $\hat{\cal M}=g_{2}[\Phi_{0}^{2}({\bf r})+\tilde{m}({\bf r})]+g_{3}[n_{c}^{2}({\bf r})+3\Phi_{0}^{2}({\bf r})\tilde{n}({\bf r})+3\Phi_{0}^{2}({\bf r})\tilde{m}({\bf r})]$, $n({\bf r},{\bf r^{\prime}})=\Phi_{0}^{*}({\bf r^{\prime}})\Phi_{0}({\bf r})+\tilde{n}({\bf r},{\bf r^{\prime}})$ and $\bar{m}({\bf r},{\bf r^{\prime}})=\Phi_{0}({\bf r^{\prime}})\Phi_{0}({\bf r})+\tilde{m}({\bf r},{\bf r^{\prime}})$.

Equations (2) and (3) describe the collective excitations of the system. The normal and the anomalous one-body density matrices can be obtained employing the transformation $\hat{\bar{\psi}}=\sum_{k}[u_{k}({\bf r})\hat{b}_{k}+v_{k}^{*}({\bf r})\hat{b}_{k}^{\dagger}]$

[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 noncondensed and anomalous densities can simply be obtained by setting, respectively $\tilde{n}({\bf r})=\tilde{n}({\bf r,r})$ and $\tilde{m}({\bf r})=\tilde{m}({\bf r,r})$ in Eqs.(4) and (5).

From now on we assume that $\tilde{n}({\bf r},{\bf r^{\prime}})=\tilde{m}({\bf r},{\bf r^{\prime}})=0$ for ${\bf r}\neq{\bf r^{\prime}}$ Bon ; Bon1 . It is worth stressing that the omission of the long-range exchange term does not preclude the stability of the system Bon ; He ; Biss ; Zhan ; Bail ; Tick ; BoudjDp .

As is well known, the full HFB theory sustains some hindrances notably the appearence of an unphysical gap in the excitation spectrum and the divergence of the anomlaous density. In fact, this violation of the conservation laws in the HFB theory is due to the inclusion of the anomalous density which in general leads to a double counting of the interaction effects. The common way to circumvent this problem is to neglect $\tilde{m}$ in the above equations, which restores the symmetry and hence leads to a gapless theory, but this is nothing else than the Popov approximation. To go consistently beyond the Popov theory, one should renormalize the coupling constant taking into account many-body corrections for scattering between the condensed atoms on one hand and the condensed and thermal atoms on the other. Following the procedure outlined in Refs Davis ; Morgan ; Boudj8 ; Boudj9 ; Boudjbook we obtain

[TABLE]

This spatially dependent effective interaction, $g_{R}$ is somehow equivalent to the many body $T$-matrix Hut . A detailed derivation of $g_{R}$, including the term $g_{3}$, will be given elsewhere. It is easy to check that if one substitutes (6) in the HFB equations, we therefore, reinstate the gaplessness of the spectrum and the convergence of the anomalous density.

III Numerical results

For numerical purposes, it is useful to set the Eqs.(1)-(6) into a dimensionless from. We introduce the following dimensionless parameters: the relative strength $\epsilon_{dd}=C_{dd}/3g_{2}$ ($\epsilon_{dd}=0.16$ for Cr atoms) which describes the interplay between the DDI and short-range interactions, and $\bar{g}_{3}=g_{3}n_{c}/g_{2}$ describes the ratio between the two-body interactions and TBI. Throughout the paper, we express lengths and energies in terms of the transverse harmonic oscillator length $l_{0}=\sqrt{\hbar/m\omega_{\rho}}$ and the trap energy $\hbar\omega_{\rho}$, respectively.

Figure.1 shows that the noncondensed and the anomalous densities increase with the TBI which leads to reduce the condensed density. A careful observation of the same figure reveals that the $\tilde{m}$ is larger than $\tilde{n}$ at low temperature which is in fact natural since the anomlaous density itself arises and grows with interactions Boudj2011 ; Boudj2012 . When the temperature approaches to the transition, one can expect that $\tilde{m}$ vanishes similar to the case of a BEC with a pure contact interaction Boudj2011 ; Boudj2012 .

In Fig.2 we compare our prediction for the condensed fraction $N_{c}/N$ with the HFB-Popov theoretical treatment and the noninteracting gas. As is clearly seen, our results diverge from those of the previous approximations due to the effects of the TBI. This means that both the condensed fraction and the transition temperature decrease with increasing the TBI.

Before leaving this section, let us unveil the role of the TBI on the collective excitations. According to Fig.3, we can observe that our results deviate from the HFB-Popov (which have also been found in Bon ) at higher temperatures for $m=0$ and $2$ excitations. The reason of such a downward shift which enlarges as the temperature approaches $T_{c}$, is the inclusion of both anomalous pair correlations and TBI. A similar behavior holds in the case of BEC with short-range interactions (see e.g. Hut ). Fig.3 depicts also that both the full HFB and the HFB-Popov produce a small shift from 1 for the Kohn mode $\omega/\omega_{\rho}=1$ at higher temperatures. One possibility to fix this problem might be the inclusion of the dynamics of the noncondensed and the anomalous components. A suitable formalism to explore such a dynamics is the time-dependent HFB theory Boudj8 ; Boudj9 ; Boudjbook ; Boudj2011 ; Boudj2012 .

IV Conclusion

In conclusion, we have deeply investigated the properties of dipolar Bose gas confined by a cylindrically symmetric harmonic trapping potential in the presence of TBI at finite temperature. The numerical simulation of the full HFB model emphasized that the condensed fraction and the transition temperature are reduced by the TBI. Effects of the TBI and temperature on the collective modes of the system are notably highlighted. We found that the full HFB approach in the presence of the TBI reproduces the HFB-Popov results of Bon only at low temperature while both approaches diverge each other when the temperature is close to $T_{c}$. One can expect that the same behavior persists in the case of a density-oscillating ground states known as a biconcave state predicted in Ref Bon1 .

Acknowledgments

We are grateful to Dmitry Petrov and Axel Pelster for the careful reading of the manuscript and helpful comments.