Rydberg atoms, with their strong dipole-dipole interactions (DDI) and dipole blockade mechanism, are highly promising for quantum information processing applications such as quantum simulators, quantum logic gates, and single-photon sources. Electromagnetically induced transparency (EIT) offers a means to significantly increase light-matter interaction time by slowing the speed of light in a medium. Combining Rydberg atoms with EIT thus facilitates substantial photon-photon interactions. Most previous studies focused on the strong-interaction, dipole blockade regime with high principal quantum numbers (n ≈ 100). This work explores an alternative approach, utilizing a high optical depth (OD) EIT medium and Rydberg atoms with a low principal quantum number (n = 32) to create a weakly-interacting many-body system. This regime, where the blockade radius is significantly smaller than the inter-particle distance, allows for the study of many-body phenomena, such as Bose-Einstein condensation (BEC), under an OD-enhanced interaction time, facilitated by the long interaction time provided by the high-OD EIT medium. This contrasts with the strongly correlated photon regimes previously investigated. The Rydberg EIT polaritons, which are bosonic quasiparticles representing superpositions of photons and Rydberg coherences, offer a unique platform for exploring this weakly-interacting many-body physics. The long interaction time enabled by high OD and the µm²-size collision cross section due to DDI between Rydberg polaritons are key features that make this system particularly promising for BEC studies.

Extensive research has explored the strong-interaction regime of Rydberg atoms and EIT, focusing on phenomena like dipole blockade and the creation of two-body photon-photon gates and strongly correlated many-body phases. However, the weakly-interacting regime, where many-body effects still manifest despite weaker individual interactions, remains less explored. The concept of dark-state polaritons, quasiparticles representing superpositions of photons and Rydberg coherences, within the EIT framework has been established, and their potential for BEC has been theoretically proposed. This work builds upon these existing theoretical frameworks and experimental demonstrations of EIT and slow light, aiming to experimentally demonstrate a weakly interacting many-body system utilizing Rydberg polaritons and leveraging the advantages of high OD to achieve significant interaction time.

The experiment was conducted using cold ⁸⁷Rb atoms in a magneto-optical trap (MOT), achieving an OD of 81 ± 3. A dark MOT was employed to increase the OD. The EIT system utilized the |5S₁/₂, F = 2, mF = 2⟩ ground state, the 32D₅/₂, |mJ = 5/2⟩ Rydberg state, and the |5P₃/₂, F = 3, mF = 3⟩ excited state. Probe and coupling fields counter-propagated, minimizing Doppler effects. A beat-note interferometer measured the phase shift of the output probe field. A mean-field model based on the nearest-neighbor distribution was developed to describe the DDI-induced attenuation and phase shift. The experimental setup included an acousto-optic modulator (AOM) to shape the probe pulse, a polarization-maintained fiber (PMF) to deliver the probe field, and a photomultiplier tube (PMT) to detect the output probe field. The transverse momentum distribution of Rydberg polaritons was measured using an electron-multiplying charge-coupled device (EMCCD) camera, allowing the observation of the cooling effect by measuring the change of beam size. The two-photon detuning was carefully determined for zero phase shift via the beat-note interferometer to avoid any lensing effect on beam profile measurement. The elastic collision rate was estimated using a theoretical formula involving the phase shift rate and the phase shift per collision. A series of experiments were conducted to measure the DDI-induced attenuation and phase shift as functions of probe Rabi frequency and coupling detuning, observing the predicted asymmetries. The cooling effect was investigated by varying the DDI strength and analyzing changes in the transverse momentum distribution of the Rydberg polaritons. The effective transverse temperature was estimated using the measured momentum distributions.

The study successfully demonstrated the DDI effect in a truly weakly-interacting regime (rg/ra < 0.1), clearly observing DDI-induced attenuation and phase shift even at rg/ra = 0.02. Experimental data on attenuation and phase shift as functions of probe Rabi frequency and coupling detuning were consistent with the developed mean-field theoretical model. A significant cooling effect was observed through a reduction in the width of the transverse momentum distribution of Rydberg polaritons as DDI strength increased. This narrowing was attributed to elastic collisions among the polaritons, driven by the high collision rate and long interaction time achieved in the system. The measured elastic collision rate was estimated to be approximately 6.0 MHz, supporting the feasibility of observing thermalization effects. The effective transverse temperature was estimated to decrease from 3.1 µK to 1.2 µK as the Rydberg polariton density increased from 1x10⁹ cm⁻³ to 2x10⁹ cm⁻³. The observed cooling effect was not solely attributed to elastic collisions but also to the EIT bandwidth, which preferentially dissipates higher-energy polaritons. The results suggest that the BEC of Rydberg polaritons is feasible, given the appropriate experimental conditions.

The successful observation of DDI-induced effects in the weakly-interacting regime demonstrates a significant advancement in the study of Rydberg polaritons. The agreement between experimental data and the mean-field theoretical model validates the theoretical approach and provides confidence in the understanding of the system's dynamics. The observed cooling effect provides strong evidence for the occurrence of thermalization processes, driven by both elastic collisions and the EIT bandwidth. These findings strongly support the feasibility of achieving Bose-Einstein condensation (BEC) of Rydberg polaritons, opening new avenues for exploring many-body physics using this unique platform. The high OD and low decoherence rate offer advantages over other polariton systems, potentially leading to longer-lived condensates and providing new opportunities to study many-body phenomena.

This research successfully demonstrated a weakly interacting many-body system of Rydberg polaritons using a high-OD EIT medium and low-n Rydberg atoms. The observed DDI-induced effects, coupled with the cooling effect, strongly suggests the feasibility of achieving BEC of Rydberg polaritons. Future research could focus on achieving stationary Rydberg polaritons, further increasing the OD, and implementing an artificial trap to enhance the likelihood of BEC. The low decoherence rate in this system offers a significant advantage for creating long-lived polariton condensates and investigating many-body physics.

The current study is limited to a two-dimensional system, where the Rydberg polaritons propagate along the longitudinal direction. Achieving a three-dimensional BEC requires additional techniques to control the longitudinal momentum. The mean-field model used might not fully capture complex many-body interactions, necessitating more sophisticated theoretical models for a more complete understanding. While the cooling effect was observed, the temperature achieved was still significantly higher than the estimated critical temperature for BEC, implying further cooling techniques are needed for BEC realization.