Spectroscopy of dipolar fermions in layered two-dimensional and three-dimensional lattices

Motivated by ongoing measurements at JILA, we calculate the recoil-free spectra of dipolar interacting fermions, for example ultracold heteronuclear molecules, in a one-dimensional lattice of two-dimensional layers or “pancakes,” spectroscopically probing transitions between different internal (e.g., rotational) states. We additionally incorporate $p$-wave interactions and losses, which are important for reactive molecules such as KRb. Moreover, we consider other sources of spectral broadening: interaction-induced quasiparticle lifetimes and the different polarizabilities of the rotational states used for the spectroscopy. Although our main focus is molecules, some of the calculations are also useful for optical lattice atomic clocks. For example, understanding the $p$-wave shifts between identical fermions and small dipolar interactions coming from the excited clock state is necessary to reach future precision goals. Finally, we consider the spectra in a deep three-dimensional lattice and show how they give a great deal of information about static correlation functions, including all the moments of the density correlations between nearby sites. The range of correlations measurable depends on spectroscopic resolution and the dipole moment.

Figure 1

Schematic of the experimental geometry considered. Heteronuclear dipolar molecules are trapped in a one-dimensional lattice of two-dimensional pancakes, with the lattice periodic in the $z$ direction. The angle $\theta$ is the angle between the intermolecule separation $\mathit{r}-{\mathit{r}}^{\prime }$ and the molecular dipole (applied electric field axis), here taken to be along the $z$ axis.

Figure 2

Lowest rotational energy levels of a single molecule in an electric field. These are described by the rigid rotor Hamiltonian in an electric field $H=B{N}^2-d_{0}E$, with $N$ the angular momentum operator, $B$ the rotational constant, and $d_0$ the dipole moment along $z$. The label $|M,m_z\rangle$ indicates the state that is adiabatically connected to the $E=0$ state with angular momentum $M$ and angular momentum $z$-component projection $m_z$. However, the $E$ field mixes states with the same $m_z$ but with different $M$, so $M$ is not a good quantum number. The energy splitting $\nu$ increases with increasing electric field and is a large energy scale even for small $E$: interactions, the trapping potential, and the kinetic energy cannot mix in states that are off-resonant bythis energy scale.

Figure 3

(a) Recoil-free spectrum with a rough modeling of the JILA experimental parameters (see text). Here $N_e\left(\omega \right)/N_e\left({\omega }_{\text{max}}\right)$ is the number of excited particles after a probe pulse of frequency $\omega$ and duration $t$ rescaled by the peak's maximum value. The overall frequency scale, related to the dipolar interaction strengths, is ${\omega }_{\text{scale}}=({\gamma }_{12}-{\gamma }_{11}+\eta )\sqrt{\frac{\pi m^3}{{\beta }^3}}$. The shape of this spectrum is entirely determined by the density profile. The infinite system has a logarithmically divergent peak at $\omega =0$, but this is rounded off by a small spectral broadening $\Gamma =7\times {10}^{-4}{\omega }_{\text{scale}}$ added to the calculation here and, less so, by the finiteness of the present system, assumed to be 200 lattice sites wide. (b) Recoil-free spectrum at $T=0$ (deeply degenerate) with ${\omega }_{\text{deg}}\equiv {\left\{m^3{\left[\frac{1024}{45}({\gamma }_{12}-{\gamma }_{11}+\eta )\right]}^2\right\}}^{-1}$, where we have chosen ${\mu }_0=2.6{\omega }_{\text{deg}}$; different ${\mu }_0$'s just rescale the plot.

Figure 4

The characteristic frequency of the collisional shifts in the thermal gas, ${\omega }_{\text{scale}}$, for transitions between KRb's two lowest rotational states vs the dipole moment $d_{11}$ in units of the permanent dipole moment $d_p$ (roughly 0.5 D for KRb). Middle curve: total characteristic frequency ${\omega }_{\text{scale}}={\omega }_{\text{sf}}+{\omega }_{\text{dir}}$. Top curve: ${\omega }_{\text{sf}}$, the contribution from the spin-flip term. Bottom curve: ${\omega }_{\text{dir}}$, the contribution from the direct term.

Figure 5

Polarizability difference broadening in the spectra. (a) Polarizability difference broadening when other shifts are negligible (as applies in a very dilute gas, or in the tails of the cloud). (b,c) Broadening including interactions and with the polarizability difference for positive (b) and negative (c) ${\omega }_2^2-{\omega }_1^2$, with magnitude decreased fivefold from its true value. They compare the thermal spectra without broadening [blue, dashed, exactly as Fig. 3a] with the spectra with broadening (red, solid) for dipole moments giving the maximal values of the collisional shifts relevant for the JILA parameters computed previously.

Figure 6

Zero-tunneling recoil-free spectra, illustrating how to read off correlations. (a) Recoil-free spectra with nearest-neighbor interactions only. The correspondence between spectral shift and spatial correlations is labeled, where the peak labeled $\left(n_1{n}_2{n}_3\right)$ has a spectral weight that is a sum over all configurations and sites $i$ with $n_1$ particles on site $i$, $n_2$ in $i$'s nearest-neighbor sites, and $n_3$ in the next-nearest-neighbor sites, summed over all sites $i$. The “$\cdots$” indicates further correlations are irrelevant to the shift, and, independent of their values at longer range, all states contribute to the peak. Here, $Z_1$ and $Z_2$ indicate the number of nearest- and next-nearest-neighbor sites; note the maximum number of atoms on (next)-nearest-neighbor sites is $Z_1$ ($Z_2$). (b) Recoil-free spectra with nearest- and next-nearest-neighbor interactions. Longer-range interactions will continue to split the peaks, and with increasing spectral resolution, longer-range correlations can be read off the spectrum. Peak heights are drawn arbitrarily.