Controllable vacuum-induced diffraction of matter-wave superradiance using an all-optical dispersive cavity

Cavity quantum electrodynamics (CQED) has played a central role in demonstrating the fundamental principles of the quantum world, and in particular those of atom-light interactions. Developing fast, dynamical and non-mechanical control over a CQED system is particularly desirable for controlling atomic dynamics and building future quantum networks at high speed. However conventional mirrors do not allow for such flexible and fast controls over their coupling to intracavity atoms mediated by photons. Here we theoretically investigate a novel all-optical CQED system composed of a binary Bose-Einstein condensate (BEC) sandwiched by two atomic ensembles. The highly tunable atomic dispersion of the CQED system enables the medium to act as a versatile, all-optically controlled atomic mirror that can be employed to manipulate the vacuum-induced diffraction of matter-wave superradiance. Our study illustrates a innovative all-optical element of atomtroics and sheds new light on controlling light-matter interactions.

the relevant Lithium transition is estimated to be around 1 − 10W/cm 2 depending on ∆t, and it is achievable with commercial laser systems.

Optical Bloch Equation.
In the presence of counter-propagating coupling fields Ω ± c and the superradiant fields Ω ± , the dynamics of the EIT mirror is described by the optical-Bloch equation which reads [S1-S5]: where Γ is the spontaneous decay rate of the excited state |3 and ρ ij is the element of the density matrix. In equations ( The evolution of the condensate densities in momentum space are shown in panels a and b where the high odd kp modes are generated in ψg while the even kp modes are generated in ψe. Panel c shows the optical coherence σeg. The coherent transfer of the BEC particle number from states |e (green line) to |g (blue solid line) is shown. Panels e-h represent the time-of-fight simulation after the condensate is released from the cavity and harmonic trap. Panels e-g are the density components carry the momenta ±kp, ±3kp, and ±5kp, respectively. The total density profile |ψg| 2 at tTOF = 1.8 ms. The contrast of the 2D plots is adjusted for better visualization.
Dynamics of BEC. When the condensate is loaded in the dispersive cavity, in the presence of the interaction between the circulating SR fields and condensate, the condensate wave functions ψ g and ψ e constitute the superposition of discrete ±(2n + 1)k p and ±2nk p plane waves, respectively. In this manner the condensate wave function can be decomposed as : where ψ (n) e,g are slowly-varying profiles. As depicted in Fig. S2 a and b, the momentum-space density profile of single realization of d opt = 500 and Ω c = 2Γ shows the clear generation of these discrete ±nk p modes. Furthermore the Fourier transform of the coherence σ eg depicted in Fig. S2 c shows clear superposition of ±nk p modes. The coherent SR-BEC interaction results in the Rabi oscillation in short time scale and transfers most of the atoms to the ground state while the rest of them decay to other states due to the incoherent spontaneous processes as shown in Fig. S2 d. The Rabi oscillation is the consequence of the strong atom-light coupling.
The generation of the highest ±nk p mode is limited by the competition between the linewidth of the superradiance and the recoil-induced Doppler shift which is given by ∆ω n ≈ Γ SR where ∆ω n = ± nk p ω/(mc) is the Doppler shift of n-th mode with ω the transition frequency and c the speed of light. In order to generate the harmonics as high as possible, one can use a heavier atom, e.g., 87 Rb, with lower recoil velocity.
After passing through the cavity, the dynamics of the condensate is described by the single component Gross-Pitaevskii equation which reads where we neglect the cross-species and atom-light interactions. The coherent transfer of the BEC particle number from states |e (green line) to |g (blue solid line) is shown. Panels e-h represent the time-of-fight simulation after the condensate is released from the cavity and harmonic trap. Panels e-g are the density components carry the momenta ±kp, ±3kp, and ±5kp, respectively. The total density profile |ψg| 2 at tTOF = 1.5 ms. The contrast of the 2D plots is adjusted for better visualization.
During the time of flight (TOF) measurement, the condensate would split into several atomic clouds which corresponds to different ±nk p modes. To simulate the TOF dynamics, we follow the standard procedures in Refs. [S6-S8] to numerically integrated the time-dependent GPE, equation. S8. To describe the motion of the cloud with different ±nk p , the TOF simulation domain is extended to 7L where L is the simulation domain in the presence of harmonic trap.
As shown in Fig. S2 e-h, we perform the TOF simulation for ∆t TOF = 1.5 ms. In Fig. S2 e-g, the evolution of the condensed atoms carrying momenta are plotted and the velocities can be calculated from the slopes which agrees with the expected values, ± k p /m, ±3 k p /m, and ± k p /m. In Fig. S2 h, the total density distribution at t TOF = 1.5 ms is shown where six density bumps that are symmetric to the origin can be observed. The two innermost density bump pair in Fig. S2 h corresponds to the ±k p modes while the density bump pair located around z = ±0.5 mm is ±3k p modes and the outermost pair carries ±5k p momenta.