Experimental Realization of Reconfigurable Photonic Lattices in Coherent Rydberg Atomic Vapors

Abstract

We experimentally demonstrated the formation of a one-dimensional electromagnetically induced optical lattice in coherently prepared three-level ⁸⁵Rb Rydberg atomic vapors with electromagnetically induced transparency (EIT). The one-dimensional photonic lattice was optically induced by a coupling field with a spatially periodical intensity distribution deriving from the interference of two strong Gaussian beams from the same laser source (~480 nm). Under the Rydberg-EIT condition, the incident weak probe beam can feel a tunable spatially modulated susceptibility, which is verified by the controllable discrete diffraction pattern observed at the output plane of the vapor cell. This investigation not only opens the door for experimentally introducing the strong interaction between Rydberg atoms to govern the beam dynamics in photonic lattices based on atomic coherence but also provides an easily accessible periodic environment for exploring Rydberg-atom physics and related applications.

1. Introduction

Dating back to the last century, Rydberg atoms with valence electrons excited to highly lying energy levels were in the spotlight, helped by the development of high-resolution absorption spectroscopy [1]. Normally, Rydberg atoms manifest a long lifetime, large electric dipole moment, strongly enhanced interactions, large susceptibility, etc. [2]. Such exaggerated properties were employed for plenty of desired scientific research, including ultralong-range Rydberg molecules and Rydberg macro-dimer [3], single-photon applications [4,5], many-body physics [6], precision measurement [7], photon-photon interactions [8,9], and quantum information [10,11,12], to name a few.

By following the obtained remarkable achievements, the current belief is that it is essential to extend these superior features of Rydberg atoms to optical lattices [13], which can engineer the distribution of cold atoms into periodic manners. Such advantageous periodic frameworks have given rise to flourishing advancements in the study of hot topics in the physical community, including superpositions of macroscopically distinct states [14], dynamical crystal [15], single-atom-resolved interferometry [16], topological physics [17,18], and parity-time symmetry [19,20], among others. Very recently, two-dimensional Rydberg atom arrays have been constructed to demonstrate anti-blockaded effects [21] and quantum nonlinear optics [22]. The physical mechanism behind the emergence of fascinating effects in Rydberg atoms trapped by optical lattices can be attributed to the periodic modulation of susceptibility arising from light-matter interactions. Such periodic susceptibility can lead to the formation of desired photonic band structures, which may play key roles in the generation of the aforementioned exotic properties, for example, topological photonics. In principle, exciting Rydberg atoms by a laser field with a spatially distributed intensity may also introduce periodicity into the susceptibility by utilizing atomic coherence such as electromagnetically induced transparency (EIT) in multi-level atomic systems [23,24].

Actually, optically induced photonic lattices [25,26,27,28] with instantaneously and in situ reconfigurability have been widely investigated by adopting EIT in lower-exited (e.g., 5S↔5P or 5S↔5D transitions) multi-level atomic configurations in recent years. Such electromagnetically induced photonic lattices are realized in both thermal and cold atomic settings [29,30] by patterning the refractive index profile in periodic manners with the assistance of EIT, which is driven by spatially structured coupling fields. Considering that EIT can effectively control the absorption, dispersion, and Kerr-type nonlinearity, electromagnetically induced optical lattices have been exploited to reveal a variety of phenomena, such as periodic Raman gain [31], edge solitons [32], all-optical tunable devices [33,34], non-Hermitian Talbot effect [35], and dynamics of vortices in the vicinity of Dirac points [36,37], that are not easily accessible in photonic lattices with fixed band structures. These previous works have paved the way for engineering periodic susceptibility with highly excited Rydberg states.

In this article, we construct a tunable one-dimension optically induced photonic lattice in a three-level ladder-type ⁸⁵Rb EIT configuration involving a Rydberg energy level. The Rydberg-state transition is accomplished by a weak Gaussian probe beam together with a strong coupling field. The coupling field is established by first splitting the output of a laser source into two identical beams and then interfering with them at the center of the ⁸⁵Rb vapor cell. As a result, the coupling field exhibits a one-dimension periodic stripe fringe along the x-direction. When the Gaussian probe field is launched into the coupling lattice along the opposite direction of the coupling field, a discrete diffraction pattern can be obtained at the output plane by adjusting the detuning of laser fields approaching the two-photon resonance, where a Rydberg-state EIT window occurs and leads to spatially periodic modulation on the refractive index of the sample. Furthermore, we explore the evolution of induced photonic lattices by observing the output diffraction patterns under different parametric regimes. A comprehensive theoretical simulation is also given to verify the experimental observations. Our work shows that atomic vapor can also provide a platform with periodicity to carry out investigations on the Rydberg atoms. This may open a new route to study the interactions between tunable band structures and properties of Rydberg excitations, potentially broadening related researches and applications.

2. Experimental Scheme

The experimental scheme is shown in Figure 1a. The optically induced photonic lattice is established in a three-level ladder-type configuration, composed of a ground state 5S_1/2(F = 3), an intermediately-exited state 5P_3/2, and a highly excited Rydberg state 37D_5/2. A weak probe field (wavelength λ₁ ≈ 780.2 nm) together with a strong coupling field (λ₂ ≈ 480 nm) excites the valence electron to the Rydberg energy level. The experimental setup is provided in Figure 1b. The probe field E₁ (frequency ω₁, horizontal polarization) emitted by a tunable external cavity diode laser (ECDL1, UniQuanta ECL801-780) is launched into a vapor cell filled with ⁸⁵Rb atoms along the z-axis to connect the transition 5S_1/2→5P_3/2. The periodic coupling field E₂ with a structured intensity distribution (frequency ω₂, vertical polarization, Rabi frequency Ω₂) is formed by interfering with two Gaussian coupling beams from the other ECDL2 (UniQuanta ECL801-480, with a maximum output power of ~240 mW) and counter-propagating with E₁ to drive the transition 5P_3/2→37D_5/2. This counter-propagating arrangement of the two fields is designed to suppress the influence of Doppler broadening [23]. The two coupling beams are placed symmetrically with respect to the z-axis at an angle of 2θ ≈ 0.25° and meet at center of the vapor cell to establish the standing-wave field with its periodicity (along the x-direction) perpendicular to propagating direction. The periodic coupling field responsible for the modulation of the spatial refractive index is shown in Figure 1c, and its period along x is determined as d = λ₂/(2sinθ) ≈ 110 µm. The atomic vapor cell with a diameter of 2.5 cm and a length of 7 cm is packaged with μ-metal sheets (to shield the external magnetic field) and a heating tape. The heating system provides a temperature of 358 K, and the atomic density is ~6.0 × 10¹² cm⁻³.

With the frequency detuning set as Δ₁ = −100 MHz relative to the resonant transition (5S_1/2→5P_3/2) driven by the probe field, it can experience substantial absorption. The presence of the coupling field can result in an EIT window when its detuning is adjusted to satisfying Δ₁ + Δ₂ = 0. Here, the frequency detuning Δ₁ (Δ₂) is defined as the difference between the laser frequency ω₁ (ω₂) and the energy gap of the two levels coupled by field E₁ (E₂) (see Figure 1a). The output probe field is divided into two parts by a PBS, and a small portion of the beam enters a photoelectric detector to monitor the EIT spectrum, while the rest part is received by a charge-coupled-device camera via an imaging lens, which guarantees the captured images are the beam profiles at the output plane of the vapor cell. Around the EIT window, one can observe clear diffraction patterns of the incident probe beam at the rear surface of the cell, indicating the formation of the spatially distributed refractive index.

3. Results

3.1. Theoretical Model

In theory, the spatial distribution of refractive index n depends on the intensity profile of the coupling lattice. By solving the density-matrix equations for the three-level atomic system under the rotating-wave approximation, the susceptibility χ of the Rydberg-EIT medium is χ = χ′ + χ″ = (2Nμ₂₁/ε₀E₁)ρ₂₁, where ε₀ is the permittivity of vacuum, N is the atomic density, μ₂₁ and ρ₂₁ are dipole moment and the density-matrix element for transition 5S_1/2→5P_3/2, respectively, excited by the probe beam, and E₁ is the electrical-field intensity of the probe field. Term n can be written as n = n₀ + Δn with Δn = n_R + in_I (n_R and n_I are the real and imaginary parts of the refractive index, respectively). Considering n₀ = 1 is the background index of the atomic gas, the modulation of the refractive index Δn caused by Rydberg EIT can be deduced as Δn = n_R + in_I ≈ χ/2, where n_R ≈ χ′/2 = Nμ₂₁/ε₀E₁ × Re(ρ₂₁) and n_I ≈ χ″/2 = Nμ₂₁/ε₀E₁ × Im(ρ₂₁). Even though the counter-propagating setting can partially suppress the Doppler effect, it cannot be neglected due to the relatively larger frequency difference ω₁ − ω₂. For an atom with a velocity of v, it feels an upshifted frequency of ω₁v/c and a downshifted frequency of ω₂v/c when it moves towards the probe and coupling beams, respectively. Consequently, the susceptibility with the Doppler effect taken into consideration can be expressed as [23,38]:

$$\chi (v)dv=\frac{i{\left|{\mu }_{21}\right|}^2/(\hslash {\epsilon }_0)}{{\gamma }_2-i{\Delta }_1-i\frac{{\omega }_1}{c}v+\frac{{\Omega }_2{(x)}_{}^2/4}{{\gamma }_3-i({\Delta }_1+{\Delta }_2)-i({\omega }_1-{\omega }_2)v/c}}N(v)dv,$$

(1)

where the spatially distributed Rabi frequency of the coupling field is defined as Ω₂(x) = μ₃₂E₂/ħ (μ₃₂ is the dipole momentum for transition 5P_3/2→37D_5/2, and E₂ is the electrical-field intensity of the coupling field) and then Ω₂(x)² implies the intensity profile of the coupling lattice; ħ is the reduced Planck constant; N(v) = N₀exp(−v²/u²)/(uπ^1/2) is the particle number density in terms of speed distribution function with u/2^1/2 being the root-mean-square atomic velocity; the decay rate γ_i is natural widths of the intermediate |2〉 and upper states |3〉 (i = 2, 3); and c is the vacuum speed of light. Equation (1) describes the profile of susceptibility χ (v) by taking the spatial distribution of Rabi frequency |Ω₂(x)|². With the Doppler effect considered, total susceptibility χ is obtained by the integration of χ (v) on the atomic velocity v:

$$\chi ={\displaystyle {\int }_{-\infty }^{+\infty }\chi (v)}dv.$$

(2)

Based on the total susceptibility χ in Equation (2), the theoretically calculated distributed real part Re[χ] of susceptibility can be obtained as Figure 2a,b when the probe field detuning is Δ₁ = −90 MHz and Δ₁ = −110 MHz, respectively, with Δ₂ = 100 MHz. By comparing the two pictures with two-photon detuning δ = Δ₁ + Δ₂ being 10 MHz and −10 MHz, one can see the distributions of the refractive index for the two cases exhibit difference, where the peak value on the index profile for δ = 10 MHz corresponds to the valley value for δ = −10 MHz. Namely, the change in the sign of the two-photon detuning can make the induced photonic lattice shift a half lattice period along the x-axis. Such a spatial movement coincides with the dispersion curve through the EIT window, where the refractive index can become larger or smaller than the background refractive index for the positive and negative sides of δ = 0 [23]. Figure 2c,d depicts the imaginary part Im[χ] corresponding to Figure 2a,b, and one can see that Im[χ] is much smaller than the Re[χ] since the EIT can partially suppress the absorption. The spatial structure of the susceptibility regulated by two-photon detuning directly determines the output diffraction patterns of the probe beam. The probe dynamics inside such a discrete environment are governed by the Schrödinger-like paraxial propagation equation:

$$i\frac{\partial \psi (x,y,z)}{\partial z}=-\frac{1}{2k_1}(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2})\psi +\frac{k_1}{n_0}{\Delta }_n(x,y)\psi ,$$

(3)

where ψ is the envelope of the incident probe beam, z is the propagation distance, and complex Δn = n_R + in_I describes the periodic potential well “written” by the standing-wave coupling field under EIT. Figure 2e,f visually demonstrates the simulated output probe field passing through the atomic sample with a transversely discrete refractive index based on Equation (3) for the aforementioned two-photon detuning with opposite signs. The numerical results support that a weak Gaussian probe beam can be effectively discretized by the photonic lattices induced in a Rydberg-EIT atomic vapor. In addition to this, the output intensity profiles are also inversed (the positions of bright and dark fringes exchanged) for positive and negative two-photon detuning.

3.2. Experimental Results and Analysis on Manipulating Two-Photon Detuning

By experimentally setting the detuning of the coupling field at Δ₂ = 100 MHz, the injected Gaussian probe field can demonstrate discrete dynamical behaviors with two-photon detuning δ = Δ₁ − Δ₂ ranging from −30 MHz to 30 MHz. The structure of the induced photonic lattice, as well as the dynamics of the probe beam, can be manipulated by tuning the related experimental parameters due to the inherited reconfigurability from multi-level atomic systems. In the first place, the diffraction patterns of the probe beam exiting the Rydberg atomic vapor are provided in Figure 3 under different two-photon frequency detunings in the vicinity of the EIT window. Experimentally, such transmission images were captured by changing the probe detuning from −120 MHz to −70 MHz but fixing the coupling-field detuning at Δ₂ = 100 MHz. With the two-photon detuning δ evolving from −20 MHz to 0, the contrast between the bright and dark fringes first becomes larger due to the increase of the modulation degree on the refractive index, and the highest contrast occurs around δ = −10 MHz (Figure 3b); then, the difference between the intensity peak and valley becomes smaller, and there is almost no difference at δ = 0 (Figure 3c). In the case of exact two-photon resonance, the probe field exhibits Gaussian distribution without any diffraction phenomena due to the disappeared modulation on the refractive index. Furthermore, the brightest pattern can be observed in this detuning point with the minimum absorption, where the imaginary part of the refractive index can be 0 in theory. It is worth noting that a very indistinct periodic intensity distribution can be observed because of the finite linewidth (~1 MHz) of the output from the adopted ECDLs. For the positive side of the two-photon resonance, the observed discretized patterns evolve similarly, with δ gradually adjusted near to the resonance.

The overall evolutions of the diffraction patterns on either side of the two-photon resonance are governed by the change of the spatially distributed susceptibility, which is in accordance with the dispersion curve of a typical EIT medium [39]. Further, the captured patterns for two-photon detuning being negative (Figure 3a,b) and positive (Figure 3d,f) possess the same period but a half-period shift of the intensity peak (bright fringes). By comparing the output at δ = −10 MHz (Figure 3b) and δ = 10 MHz (Figure 3d), one can see that the shift of the bright fringes advocated the prediction in theoretical Figure 2e,f.

3.3. Experimental Results and Analysis on Manipulating Coupling Power

In order to achieve high diffractive efficiency in an electromagnetically induced lattice, the constructive and destructive interference should be carefully elaborated to realize the demanding spatial distribution of Rabi frequency with enough difference between maximum and minimum values according to Equation (1). Here, the diffraction efficiency of the + nth-order diffractive beam is defined as η_n = I_n′/I₀, where I_n′ represents the intensity of a selected + nth-order diffractive beam (when n = 0, I_n′ represents the intensity of 0th-order diffraction beam) and I₀ is the intensity of the probe beam transmitting through vapor without periodical refractive index [33,40]. Considering the large dipole momentum of a Rydberg atom, its Rabi frequency can be much smaller compared to lower-excited atoms under the same pump power. In other words, the high-contrast distribution of Rabi frequency in Rydberg atoms can be realized only for high enough coupling power. Figure 4 demonstrates the dependence of diffractive efficiency (reflected by the intensity contrast of the output patterns) of the launched probe beam on the power of the coupling lattice. The two-photon detuning is chosen as 10 MHz to ensure the maximum contrast of the diffraction patterns. Here, the power means the total power of the two identical coupling beams responsible for the generation of the standing-wave coupling field. With the growth of the coupling power, it is clear that the diffractive patterns become more obvious, indicating an effective enhancement of the contrast between the peak and valley of the refractive index as well as the Rabi frequency. A larger coupling-field Rabi frequency means the excitation from intermediate states to Rydberg states can be more efficient. According to the observations, a clear diffraction pattern occurs when the coupling power increases over 120 mW, which is much larger than the required ~30 mW for the similar ladder-type EIT system but with a lower excitation (5P_3/2→5D_5/2) [29]. With the power of the coupling field increased to 240 mW, the diffractive efficiencies reach the maximum with η₀ ≈ 20%, η₁ ≈ 16%, and η₂ ≈ 7%, corresponding to 0th-, +1st-, and +2nd-order diffraction beams, respectively.

4. Discussion and Conclusions

In the present work, we experimentally realized the establishment of a one-dimensional electromagnetically induced optical lattice (with its period along the x-axis) in a three-level ladder-type ⁸⁵Rb atomic vapor with a Rydberg excitation. The strong coupling field responsible for the periodic modulation of the refractive index counter-propagates with a weak probe field to reduce the influence of Doppler broadening, and thus a Rydberg EIT window occurs. At the output plane of the vapor cell, clear one-dimensional discrete diffraction patterns were observed near the two-photon resonance, supporting the formation of periodic susceptibility in such a thermal Rydberg-EIT atomic configuration. The dynamical evolution of diffraction patterns through manipulating experimental parameters indicates the easily accessible reconfigurability and controllability of the established photonic lattices. The unique property of the Rydberg-EIT-based photonic lattice is that the strong Rydberg-Rydberg interactions are introduced as an indispensable degree of freedom for manipulating the behaviors of light in periodic structures.

Compared to the optically induced photonic lattices based on EIT in lower-excited multi-level atoms, the introduction of strong Rydberg–Rydberg interactions [41] may be promising for operating nonlinear regimes as well as a variety of interesting proposals in future experiments, such as nonlinear light diffraction [42], parity-time symmetry optics [19], nonlocal optical solitons [20], single light bullets [43], and spatiotemporal solitons [44]. However, the experimental demonstration of photonic lattices with strong nonlinearity is lacking. The current optically induced photonic lattice with giant nonlinearity [45] can certainly provide a fertile ground for exploring dynamical behaviors of lights dominated by nonlinearity in periodic structures. Furthermore, Rydberg atoms trapped in optical lattices have spawned flourishing achievements in demonstrating unique optical features [14,15,16,17,18,19] by taking advantage of the interactions between Rydberg atoms and periodic environments. The currently demonstrated electromagnetically induced photonic lattice in coherently prepared atomic vapors, which possesses controllable Rydberg–Rydberg interactions and periodicity simultaneously but avoids the sophisticated laser-cooling equipment for trapping atoms into optical lattices, may pave the way to study the interactions between Rydberg atoms and periodicity in room temperature.