Interferometry of Efimov states in thermal gases by modulated magnetic fields

We demonstrate that an interferometer based on modulated magnetic field pulses enables precise characterization of the energies and lifetimes of Efimov trimers irrespective of the magnitude and sign of the interactions in ${}^{85}\mathrm{Rb}$ thermal gases. Despite thermal effects, interference fringes develop when the dark time between the pulses is varied. This enables the selective excitation of coherent superpositions of trimer, dimer, and free-atom states. The interference patterns possess two distinct damping timescales at short and long dark times that are either equal to or twice as long as the lifetime of Efimov trimers, respectively. Specifically, this behavior at long dark times provides an interpretation of the unusually large damping timescales reported in a recent experiment with ${}^7\mathrm{Li}$ thermal gases [Yudkin et al., Phys. Rev. Lett. 122, 200402 (2019)]. Apart from that, our results constitute a stepping stone towards a high precision few-body state interferometry for dense quantum gases.

Figure 1

(a) Energy spectrum of three harmonically trapped ${}^{85}\mathrm{Rb}$ particles with ${\omega }_r/\left(2\pi \right)=350\mathrm{Hz}$. Efimov trimer (T), atom-dimer (AD), and trap (A) states are depicted. Initially, the scattering length is set at $a_{\text{bg}}=819a_0$ (dashed vertical line), then modulated with amplitude $a_m$ (gray region). Note $a_0$ is the Bohr radius. (b) A schematic illustration of the Ramsey-type interferometer: A first pulse with envelope $\chi \left(t\right)$ associates atom dimers and Efimov states out of trap states [first and second subgraphs in (b)]; the system then evolves freely during the dark time $t_d$ [third subgraph in (b)], while a second pulse further admixes the states together with their dynamical phases that were accumulated during $t_d$ [fourth subgraph in (b)]. (c) The ratio of thermally averaged (RTA) probabilities, ${\mathbb{P}}_T\left(t_d\right)$ at $a_{\text{bg}}=819a_0$, and distinct temperatures (see legend). Inset: A zoom-out plot of RTA at early $t_d$. (d) [(e)] Frequency spectra referring to region I (II) of the RTA quantifying its single (multifrequency) behavior at different values of temperature $\mathcal{T}$. The vertical dotted lines correspond to the three-level model (TLM) predictions for $E_T^{\left(2\right)}, E_{AD}^{\left(1\right)}$, and a trap state (see text).

Figure 2

(a) ${\mathbb{P}}_T\left(t_d\right)$ for different temperatures (see legend), taking into account the decay width, ${\mathrm{\Gamma }}^{\left(2\right)}/h=748\mathrm{Hz}$, of the first excited Efimov state at $a_{\text{bg}}=2030a_0$. The gray solid lines outline the upper and lower peak envelopes. The inset presents the frequency spectrum pertaining to region II, $|F_{\text{II}}\left(\omega \right)|$. (b) The mean peak-to-peak envelope, ${\mathbb{P}}_T^p\left(t_d\right)$, is fitted with the exponentials $f_{i/ii}\left(t_d\right)=g_{i/ii}{e}^{-{\mathrm{\Gamma }}_{i/ii}(t_d-t_{i/ii}^0)/\hslash }+w_{i/ii}$ at dark time intervals $i$ and $ii$ (black and green dashed lines) with $g_{i/ii}, w_{i/ii}$ representing fitting constants. The characteristic decay time of the oscillations at long $t_d$ is twice as long as the intrinsic Efimov lifetime $\hslash /{\mathrm{\Gamma }}^{\left(2\right)}$.

Figure 3

(a) ${\mathbb{P}}_T\left(t_d\right)$ at $a_{\text{bg}}=-2030a_0$ and various temperatures (see legend). The driving frequency is resonant with the transition between the ground Efimov and the first trap state, and the decay width of the former ${\mathrm{\Gamma }}^{\left(1\right)}/h=41\mathrm{kHz}$. The inset presents the frequency spectrum of region I, $|F_I\left(\omega \right)|$. (b) The mean peak-to-peak envelope, ${\mathbb{P}}_T^p\left(t_d\right)$ at $\mathcal{T}=270\mathrm{nK}$ is fitted with $f_{i/ii}\left(t_d\right)=g_{i/ii}{e}^{-{\mathrm{\Gamma }}_{i/ii}(t_d-t_{i/ii}^0)/\hslash }+w_{i/ii}$ at the dark time intervals $i$ and $ii$. Even at attractive interactions, the energy and lifetime of Efimov states can be simultaneously assessed.