The phenomenon of electromagnetically induced transparency (EIT), which was first observed in atomic vapors by Harris and coworkers1,2,3, has attracted much attention of the physics community over the last few decades. One of the most fascinating features of EIT has been the realization of slow light4where the group velocity of light becomes much lower than that in the vacuum and has important applications in optical telecommunication and interferometry5,6. More recently, it has been shown that optomechanical cavities with moving mirrors can be used as another platform for manifestation of EIT which is called optomechanically induced transparency (OMIT)7,8,9,10,11,12,13,14,15,16,17,18.

In a bare optomechanical system (OMS), where the moving end mirror or the membrane in the middle acts as a mechanical oscillator, the radiation pressure of the cavity optical field is coupled with the vibrational mode of the mechanical oscillator through the optomechanical coupling19,20,21. In parallel, there are hybrid optomechanical systems which consist of cavities containing atomic systems like an ensemble of ultracold atoms or a Bose–Einstein condensate (BEC) where the collective mode excitation of the atomic field plays the role of the mechanical oscillator which couples to the radiation pressure of the cavity22,23,24. One of the most interesting features of such hybrid systems containing BEC is the nonlinear effect of the atom-atom interaction which plays the role of an atomic parametric amplifier25,26,27,28,29,30,31,32. It has been shown that the nonlinear effect of atomic collisions may be used for the generation of Casimir photons33, strong quadrature squeezing34, and ultraprecision quantum sensing35,36,37.

Ultraslow light realization using an interacting Bose–Einstein condensate trapped in a shallow optical lattice

In recent years, optomechanical systems have been proposed as suitable candidates to substitute for atomic gases in order to generate slow and fast light based on the phenomenon of OMIT38,39,40. Besides, it has been shown that hybrid optomechanical systems consisting of atomic systems or BEC may provide more controllability in fast and slow light realization41,42,43,44,45,46,47. Furthermore, the phenomenon of Fano resonance which was observed for the first time in some of the Rydberg spectral atomic lines, can be also observed in optomechanical systems48,49,50.

In the study of ultracold atoms or BECs trapped inside periodic potentials of optical lattices, there are two important regimes of deep and shallow optical lattice where the system is described based on different mathematical models51. In the regime of deep optical lattice, which is also referred to as the tight-binding limit, the system is described by the Bose–Hubbard model51where the atomic field is expanded in terms of the Wannier wave functions. It has been shown52that in a system consisting of a BEC inside a deep optical lattice which obeys the Bose-Hubbard model, the time delay of the slow light group velocity can be increased up to about \(55\,\upmu\mathrm{s}\) in the best conditions by controlling the effective cavity detuning, the pumping rate of the cavity, and the strength of atomic collisions. On the other hand, for obtaining slow lights with longer time delays, it is necessary to decrease the depth of the optical lattice by decreasing the pumping rate of the cavity. However, in the limit of shallow optical lattice, the Bose-Hubbard model loses its validity and the system should be described by a model in which the atomic wave function is expanded in terms of plane waves and can be considered as a single-mode quantum field in the Bogoliubov approximation53,54. It has been shown55that in a hybrid optomechanical system consisting of a BEC trapped in a shallow optical lattice, the time delay of the slow light can be enhanced up to 0.8 ms by decreasing the depth of the optical lattice in the absence of atom-atom interaction.

Motivated by the above-mentioned investigations on slow light realization, in this article we propose an experimentally feasible scheme for the ultraslow light realization based on the OMIT phenomenon using a hybrid optomechanical system consisting of a one-dimensional BEC trapped in a shallow optical lattice considering the nonlinear effect of atom-atom interaction. It is assumed that the cavity is pumped by a coupling laser, which is responsible for the generation of the optical lattice and is tuned at the red sideband of the cavity effective frequency, and simultaneously is probed by another laser with a much weaker pumping rate. It is shown that at a fixed value of the coupling laser pumping rate, one can make the system change its regime from the normal mode splitting (NMS) to the OMIT by increasing the s-wave scattering frequency of atomic collisions. Then, we show that an ultraslow light with a time delay more than 150 ms, which corresponds to a group velocity as low as 1 mm/s, is achievable by decreasing the optical lattice depth through decreasing the pumping rate of the cavity and also by increasing the strength of atom-atom interaction through the transverse trapping frequency of the BEC and controlling the number of the atoms of the BEC. We also explicitly show that the mentioned time delay occurs in the frequency region of coupling-probe detuning where the reflection coefficient of the cavity is maximum and therefore an ultraslow light with a considerable amplitude is physically observable in the output field of the cavity.

The paper has been organized as follows: In “System Hamiltonian” section the Hamiltonian of the system is described and in “Dynamics of the system” section the dynamics of the system is modeled based on the Heisenberg-Langevin equations. In “Results and discussion” section it is shown how an ultraslow light with a considerable time delay can be realized in the output of the cavity based on the OMIT phenomenon. Finally, the summary and conclusions are given in “Summary and conclusions” section.