Ion-induced interactions in a Tomonaga-Luttinger liquid

We investigate the physics of a Tomonaga-Luttinger liquid of spin-polarized fermions superimposed on an ion chain. This compound system features (attractive) long-range interspecies interactions. By means of density matrix renormalization group techniques we compute the Tomonaga-Luttinger-liquid parameter and speed of sound as a function of the relative atom/ion density and the two quantum defect parameters, namely, the even and odd short-range phases which characterize the short-range part of the atom-ion polarization potential. The presence of ions is found to allow critical tuning of the atom-atom interaction, and the properties of the system are found to depend significantly on the short-range phases due to the atom-ion interaction. These latter dependencies can be controlled, for instance, by manipulating the ions' internal state. This allows modification of the static properties of the quantum liquid via external driving of the ionic impurities.

Figure 1

Sketch of the physical system considered in this work. A linear ion crystal, whose ions are positively charged (big blue spheres) and separated by a distance $D$, and a Tomonaga-Luttinger liquid of ultracold atoms (indicated by the red cloud with small spheres) that overlaps the crystal.

Figure 2

A diagram of the effective potential generated by two ions. The black line shows the total potential, in this case a box with two ions as described by the model potential, whereas the blue horizontal lines indicate the energy of a specific eigenstate. A blue circle on a blue line indicates that this eigenstate has been occupied by a fermion. The number of ion-bound states per ion depends on the model parameters. The upper diagram shows the ion-bound (IB) and trap-bound (TB) filling types for the case of one ion-bound state, the lower for the case of two ion-bound states.

Figure 3

Filling of the lower-energy states will correspond to a screening of the attractive part of the atom-ion potential. As shown on this diagram, this can be understood microscopically as an effective cancellation of the wells on either side of the ion, ultimately only leaving a soft barrier. The expected effect in TLL terms would be a raising of the value of $K$ for low filling, corresponding to induced attraction, and a lowering of the value of $K$ for high fillings, corresponding to induced repulsion.

Figure 4

Speed of sound for ion-bound states of a system of ions and interacting fermionic atoms with $N_{\text{A}}=N_{\text{I}}$. All points have errors of $\pm 0.05$. The dashed line schematically marks the transition from the weak-ion domain (above the dashed line) the strong-ion domain (below the dashed line) (see the text for further details). The ions significantly lower the speed, corresponding to a hindering of collective excitations, especially in the strong domain (below the dashed line).

Figure 5

TLL parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\text{A}}=N_{\mathrm{I}}$. All results have errors less than $\pm 0.02$.

Figure 6

Luttinger-liquid parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\text{A}}=2N_{\mathrm{I}}$. All results have errors less than $\pm 0.02$.

Figure 7

Luttinger-liquid parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\text{A}}=3N_{\mathrm{I}}-2$. All results have errors less than $\pm 0.05$.

Figure 8

An example of values of $K\left(N_s\right)$ for trap-bound systems with constant atom and ion densities, constant lattice separation, and different sizes (as measured by number of lattice points $N_s$). To calculate $K={lim}_{N_s\to \infty }K\left(N_s\right)$ we apply to finite systems and extrapolate the results to the infinite-size limit $1/N_s\to 0$ using a linear fit. The value of $K$ in the limit is shown with an error which is the $2\sigma$ confidence interval on the fit. This example has $\omega =10{\left(R^{*}\right)}^{-4}$, $\gamma ={\gamma }_{\text{min}}$, and $a=0.1R^{*}$, and $N_{\text{A}}=3N_{\mathrm{I}}-2$.