Effective two- and three-body interactions between dressed impurities in a tilted double-well potential

TL;DR

The work addresses how a bosonic bath mediates effective interactions among a small number of impurities confined in a tilted double-well within a 1D ring trap. Using the ab initio ML-MCTDHX method, the authors demonstrate that impurity density in the energetically elevated well encodes medium-induced two-body attraction and reveals the presence of three-body induced interactions, which are captured by energy-based effective models. Two- and three-body effective descriptions are constructed, with the two-body model reliably reproducing density configurations in the weak-to-intermediate coupling regime, while the three-body model provides only qualitative agreement and requires renormalization in stronger coupling. The study extends to three-component mixtures, showing the richness of mediated interactions and offering pathways to detect and tune two- and three-body induced forces in cold-atom experiments.

Abstract

We explore the impact and scaling of effective interactions between two and three impurity atoms, induced by a bosonic medium, on their density distributions. To facilitate the detection of mediated interactions, we propose a setup where impurities are trapped in a tilted double-well potential, while the medium is confined to a ring. The tilt of the potential breaks the spatial inversion symmetry allowing us to exploit the population of the energetically elevated well as a probe of induced interactions. For two impurities, the interaction with the medium reduces the impurity population at the energetically elevated well, which we interpret as evidence of induced impurity-impurity attraction. Furthermore, the impact of an induced three-body interaction is unveiled by comparing the predictions of an effective three-body model with many-body simulations. We extend our study for induced interactions to a three-component mixture containing distinguishable impurities. Our results suggest pathways to detect and tune induced two- and three-body interactions.

Figures (11)

• Figure 1: Sketch of our impurity-medium setup. (a) One-body densities of the medium $A$ (red) and the three bosonic impurities (blue) for repulsive impurity-medium interactions. The medium, which consists of $N_A=12$ weakly interacting bosonic particles, is confined to a ring with periodic boundary conditions. The impurities are trapped by a tilted double-well potential (gray line). (b) The medium particles (red circles) interact via a contact interaction of strength $g_{AA}$. The impurities (blue circles) repel each other with strength $g_{BB}$. The boson-impurity interaction is denoted by $g_{AB}$. (c) The induced interactions between the three impurities are parameterized by the effective two-body ($g_{BB}^{\mathrm{eff}}$) as well as three-body ($g_{BBB}^{\mathrm{eff}}$) effective couplings. (d) Overview of the identified scaling behavior of the mediated two- and three-body interaction strengths between two or three impurities belonging to species $B$ and $C$.
• Figure 2: Ground-state one-body densities of (a) the medium and (b) the two bosonic impurities as a function of the interspecies interaction strength $g_{AB}$. (c) Population of impurities at the energetically elevated double-well site (located at $x^B>0$), $I_{BB}^{\mathrm{MB}}$, see Eq. (\ref{['eq:int_obd']}) for the definition, with respect to $g_{AB}$. The simulations are performed within the many-body approach ML-MCTDHX. The two repulsively interacting ($g_{BB}=0.1$) impurities experience a tilted double-well potential and are coupled to a bosonic medium with $N_A=12$ and $g_{AA}=0.1$.
• Figure 3: (a) The energy $E_{BB}^{\mathrm{pol}}$ (in units of $\hbar \omega$) and the corresponding effective interaction strength $g_{BB}^{\mathrm{eff}}$ (in units of $\sqrt{\hbar^3 \omega/m}$) in terms of $g_{AB}$. The energies are obtained either using the many-body approach, $E_{BB}^{\mathrm{pol}}$, or the mean-field approximation, $E_{BB}^{\mathrm{pol, MF}}$ (see legend). Notice that the predictions of the two approaches agree well. (b) Integrated one-body density [Eq. (\ref{['eq:int_obd']})] determined via the many-body approach $I_{BB}^{\mathrm{MB}}$, the effective two-body model [Eq. (\ref{['eq:eff_ham_BB']})], $I_{BB}^{\mathrm{eff}}$, and the mean-field approximation, $I_{BB}^{\mathrm{MF}}$. The blue shaded areas in panels (a) and (b) mark the interaction region where the relative deviation between $I_{BB}^{\mathrm{MB}}$ and $I_{BB}^{\mathrm{MF}}$ is smaller than $0.1$. (c) One-body density of the impurities within the many-body approach (MB), the effective two-body model (2b,eff) and the mean-field approximation (MF), see also legend. The double-well potential is also shown and the inset illustrates the impurities' density within the right well as predicted for the different methods. Here, $g_{AB}=0.5$, see the vertical dotted lines in panels (a) and (b). Other system parameters are $N_A=12$ and $g_{AA}=g_{BB}=0.1$.
• Figure 4: (a) The energy $E_{BBB}^{\mathrm{pol}}$ and the effective three-impurity interaction strength $g_{BBB}^{\mathrm{eff}}$ calculated by fitting the effective model to $E_{BBB}^{\mathrm{pol}}$ (see main text). $E_{BBB}^{\mathrm{pol}}$ is compared to $E_{BBB}^{\mathrm{pol}, MF}$, which is computed within the mean-field approach. (b) Integrated one-body density of the impurities, $I_{BBB}$, computed within the many-body (MB) method, the mean-field (MF) approximation and the three-body effective model with and without a three-body interaction potential, labeled as (3b,eff) and (3b,eff,0), respectively. (c) Ground-state density distribution of the impurities within four different approaches (see legend) for fixed $g_{AB}=0.5$ indicated also by vertical gray dotted lines in panels (a) and (b). The inset provides a magnification of the densities for the right well of the double well (see also the right axis of panel (c)) emphasizing the degree of agreement among the different approaches. Other system parameters are $N_A=12$, $N_B=3$, $g_{AA}=0.1$ and $g_{BB}=0.1$.
• Figure 5: (a) Integrated one-body density of the impurity $B$ (in the energetically higher well) as a function of the impurity-medium coupling strengths $g_{AB}$ and $g_{AC}$. (b) Effective two-body interaction strength (subtracting $g_{BC}$) obtained from an effective two-body model [cf. Eq. (\ref{['eq:eff_ham_BB']})]. Integrated density of the (c) $B$ and (d) $C$ impurities for varying $g_{AC}$ and fixed $g_{AB}=0.2$ [see the vertical blue solid lines in panels (a) and (b)] obtained within the many-body (MB), mean-field (MF) and effective two-body model (2b,eff). The encircled parametric regions in panel (b) where $\mathcal{E}_{BC}^{\mathrm{MF}}<0.03$ (orange-white dotted line) and where $\mathcal{E}_{BC}^{\mathrm{2b,eff}}<0.03$ (gray-white dotted line) signify an extended range of validity of the effective two-body model compared to the mean-field results. The blue shaded areas in panels (c), (d) denote $\mathcal{E}_{BC}^{\mathrm{2b,eff}}>0.03$. In all panels, the system consists of a weakly-interacting bosonic ultracold gas in a ring potential coupled to two impurities $B$ and $C$ interacting with $g_{BC}=0.1$.