A weakly-interacting many-body system of Rydberg polaritons based on electromagnetically induced transparency

Rydberg-EIT systems enable strong photon-photon interactions via dipole-dipole interactions (DDI) among Rydberg excitations while EIT provides long interaction times by slowing light. Prior studies focused on the strong blockade regime with high principal quantum number n ≈ 100, enabling few-photon gates and strongly correlated phases. This work proposes and demonstrates a complementary approach: using a high optical depth (OD) EIT medium with a low-n Rydberg state to realize a weakly interacting many-body system of Rydberg polaritons, where rg/ra < 1 (even < 0.1). The research questions are whether DDI effects remain observable in this weak regime, how they manifest in attenuation and phase shift (linked to inelastic and elastic collisions), and whether elastic collisions drive thermalization evidenced by narrowing transverse momentum distributions, pointing toward feasibility of polariton Bose-Einstein condensation (BEC).

The paper situates itself within work on Rydberg blockade and quantum information (e.g., photon-photon gates, simulators), and EIT-based slow/stationary light enhancing light-matter interaction times. Strongly interacting Rydberg-EIT phenomena have demonstrated correlated photons and bound states. Theoretical frameworks for dark-state polaritons and stationary-light polaritons relate dispersion and momentum distributions to effective mass and temperature, and suggest BEC of polaritons under suitable conditions. Prior observations include nonlocal nonlinear optics and dephasing in Rydberg gases, and anisotropic interactions for well-oriented Rydberg D-states. The present work targets the weak-interaction regime, less explored experimentally, leveraging high OD to achieve sufficient interaction time so that even weak DDI produces measurable many-body effects.

Experimental system: Cold 87Rb atoms in a MOT with ~5×10^8 atoms at ~350 μK, cloud size ~1.8×1.8×6.0 mm^3. After switching off MOT fields, a dark-MOT phase of 2.5 ms increased OD to α = 81 ± 3 (adjustable to lower OD when needed). All atoms were optically pumped to |5S1/2, F=2, mF=2⟩. Probe and coupling fields were σ+ polarized, addressing |1⟩=|5S1/2, F=2, mF=2⟩ ↔ |3⟩=|5P3/2, F=3, mF=3⟩ (Γ=2π×6 MHz), and |3⟩ ↔ |2⟩=|32D5/2, mJ=5/2⟩. The Rydberg lifetime τ1 ≈ 30 μs; intrinsic decoherence Γ2 ≈ 0 here, yielding negligible intrinsic dephasing. Estimated atomic density natom ≈ 5×10^10 cm−3. For 32D5/2, mJ=5/2 with Dc=0.343, C (DDI parameter) ≈ −2π×260 MHz μm. Probe and coupling beams counter-propagated to suppress Doppler; e−1 full widths were 130 μm (probe) and 250 μm (coupling). A beat-note interferometer measured probe phase shifts and precisely determined δ=0. Probe pulses were shaped via a double-pass AOM; detection via PMT and oscilloscope for temporal data and EMCCD for transverse profiles. Laser preparation: Probe generated by homemade diode laser injection-locked to a master ECDL (780 nm) stabilized by PDH lock and saturated absorption spectroscopy. Coupling (≈482 nm) from a TA-SHG pro system, PDH-locked using FIT spectroscopy in a heated Rb vapor cell. The sum frequency of probe and coupling was stabilized to the EIT transition with RMS fluctuation ~150 kHz. Parameters α, Ωc, and γ0 were determined as in prior work; Δ was set (e.g., Δc = ±Γ, 0). Intrinsic decoherence maintained at γ0 ≈ 0.012(1) (in units of Γ) throughout. Theoretical model: A mean-field model for weakly interacting Rydberg polaritons using the nearest-neighbor distribution relates DDI-induced attenuation Δβ and phase shift Δφ to system parameters. Analytical steady-state expressions show Δβ and Δφ ∝ Ωp^2 with asymmetric dependence on single-photon detuning Δ due to the DDI sign (C6 < 0). Key parameters include SDDI = παΓC6natom/30^3, W = √(Γ^2+4Δ^2), and a phenomenological ε linking Rydberg population P22 to Ωp and Ωc. Assumptions: laser linewidths Δωc, Δωp ≪ Γ; two-photon detuning δ negligible compared to EIT linewidth for Δβ and Δφ. Phase and attenuation measurements: At δ=0 (determined via beat-note interferometer), measured steady-state attenuation coefficient β and phase shift φ as functions of Ωp^2 for several Δ. Linear fits yielded slopes χβ and χφ versus Δ, compared to theory to extract SDDI and ε. Transverse momentum distribution measurements: To probe thermalization, the output probe beam profile was imaged on an EMCCD. The probe waist at the atomic cloud was reduced to 39 μm (using lenses L4, L5, L2) to broaden initial transverse momentum; coupling beam remained large (≈6.4× probe size) to minimize waveguiding. Chosen Δc=+1Γ to keep interaction time similar while reducing DDI-induced attenuation. The two-photon detuning δ0 was set such that the total phase shift φ=0 (canceling DDI-induced φ>0 by negative δ) to avoid lensing. For each Ωc value, δ0 was determined via linear φ(δ) fits; uncertainty ±2π×30 kHz. Beam profiles were fitted with 2D Gaussians; transverse momentum distributions were derived from image intensities. Control measurements at δ0±2π×50 kHz checked lensing sensitivity; additional control at Δ=−1Γ checked for nonlinear self-focusing contributions. Collision rate estimation: Using dφ/dt = Rc φc, with dφ/dt ≈ Δφ/τd from measured phase shifts and delay time τd=2.1 μs (Δc=+1Γ, Ωp≈0.2Γ), and φc = k a (hard-sphere approximation) with blockade radius a≈2.1 μm and measured k≈0.051 μm−1, the elastic collision rate Rc was estimated.

- Demonstration of a weakly interacting many-body system of Rydberg polaritons using low-n (n=32) Rydberg state and high-OD (α≈80) EIT medium, achieving rg/ra < 0.1 and observable DDI effects even at rg/ra ≈ 0.02. - DDI-induced attenuation β and phase shift φ scale linearly with Ωp^2 and exhibit asymmetry versus single-photon detuning Δ, consistent with mean-field theory. Linear-fit slopes yielded SDDI ≈ 37 Γ^3/γ0^2 (from β) and ≈ 38 Γ^3/γ0^2 (from φ), with γ0 ≈ 0.012. From SDDI and system parameters, ε = 0.43 ± 0.05. - Careful determination of δ=0 via beat-note interferometry with uncertainty ±30 kHz ensured accurate phase/attenuation measurements. - Thermalization evidence: Increasing DDI strength (higher Ωc and polariton density) narrowed the transverse momentum distribution at the output. e−1 full widths of momentum distributions: 0.10 μm−1 (no atoms), 0.087 μm−1 (with atoms, Ωc=0.1Γ), 0.065 μm−1 (with atoms, Ωc=0.2Γ). Corresponding EMCCD beam diameters: 2.6 mm (no atoms), 2.2 mm (Ωc=0.1Γ), 1.6 mm (Ωc=0.2Γ). - Effective transverse temperature Teff decreased from ~3.1 μK (no atoms) to ~2.0 μK (Ωc=0.1Γ) to ~1.2 μK (Ωc=0.2Γ), a ≈2.6× reduction at higher DDI strength. - Collision rate estimate: With τd=2.1 μs, Δc=+1Γ, Ωp≈0.2Γ (density ≈2×10^9 cm−3), dφ/dt≈0.64 rad/μs, blockade radius a≈2.1 μm, and k≈0.051 μm−1, obtained φc≈0.11 rad and Rc≈6.0 MHz. - Control checks: Lensing effects around δ0 were minimal (beamwidth slopes vs δ: 2.5 μm/kHz at Ωc=0.1Γ; 2.1 μm/kHz at Ωc=0.2Γ). At Δ=−1Γ, where DDI-induced φ is ≈5–6× smaller and self-focusing negligible, a reduction in beam size persisted, supporting collision-driven thermalization rather than nonlinear self-focusing. - No anisotropy was observed in transverse images, consistent with random orientations of interatomic axes in the ensemble of D-state Rydberg atoms.

The results confirm that even in the weak-interaction regime (low-n, moderate densities) DDI among Rydberg polaritons produces measurable attenuation and phase shifts, validating a mean-field description based on nearest-neighbor statistics. The linear dependence of β and φ on Ωp^2 and their asymmetric detuning dependence match theoretical predictions and enable extraction of interaction parameters (SDDI, ε). The observed narrowing of the transverse momentum distribution with increasing DDI indicates elastic collisions drive thermalization and cooling in the transverse degrees of freedom within μs interaction times. Additionally, EIT bandwidth likely assists cooling by preferentially dissipating higher-momentum polaritons created in collisions. Together, the μm-scale collision cross section and μs-scale interaction times in a high-OD medium create favorable conditions for many-body phenomena, suggesting feasibility of achieving Rydberg polariton BEC when combined with stationary-light techniques and enhanced OD.

This work introduces and experimentally realizes a weakly interacting many-body system of Rydberg polaritons using a high-OD EIT medium and a low-n Rydberg state (n=32). It demonstrates clear DDI-induced attenuation and phase shifts at rg/ra well below unity and reveals collision-driven thermalization via significant narrowing of the transverse momentum distribution, corresponding to a ~2.6× reduction in effective transverse temperature. A mean-field theoretical model captures the observations and yields quantitative interaction parameters. The combination of large interaction cross section and long interaction time points to a practical pathway toward Bose-Einstein condensation of Rydberg polaritons, especially if made stationary and supported by higher OD and trapping. Future directions include realizing stationary Rydberg polaritons with extended lifetimes, further increasing OD to enhance interaction times, implementing artificial trapping potentials for evaporative cooling, and exploring both two- and three-dimensional polariton condensates and many-body phases.

- The present experiments probe transverse thermalization in a propagating (effectively two-dimensional) polariton system; the longitudinal kinetic energy (~140 mK) is much larger than the transverse temperature, precluding BEC in this configuration. - Achieving BEC will require stationary polaritons, higher OD (for longer interaction/storage times), and likely an external trap to facilitate evaporative cooling or controlled dissipation. - DDI-induced effects can become unobservable at too low Ωp or low OD; measurements rely on careful stabilization of two-photon detuning (uncertainty ±30 kHz) and control of lensing effects. - At certain detunings (e.g., Δ=−1Γ), strong attenuation limits the observable degree of cooling despite reduced self-focusing; mean-field modeling assumes weak interactions and nearest-neighbor statistics, which may not capture all many-body correlations beyond the weak regime. - Random orientations of Rydberg D-state atoms in an ensemble average out anisotropic interaction signatures in transverse images.