Phase difference induced curved photonic lattice in atomic vapors

Abstract

As an optical structure characterized by modulations in both transverse and longitudinal directions, the curved photonic lattice has been widely utilized to mimic the behaviors of electrons that are challenging to observe directly. Within the framework of quantum-optical analogies, such a mimicking approach offers additional degrees of freedom for manipulating the dynamics of light. Here, we report the realization of curved photonic lattices in atomic vapors by employing the phase difference in a two-beam interference configuration. The phase difference is introduced via an electro-optic modulator. A weak Gaussian probe field is sent into the curved photonic lattice to image the constructed structure in the context of electromagnetically induced transparency both experimentally and theoretically. In the case of sinusoidal modulation of the phase difference, the transverse oscillation of the output probe pattern is observed clearly, indicating the instantaneous and accurate control of the curved photonic lattice. It is also found that the modulation frequency can play an important role in synthesizing flat bands. This work not only presents a convenient way for producing curved photonic lattices with reconfigurability, but also puts forward a promising platform to investigate quantum-optical analogies in atomic vapors.

Graphical abstract

Keywords

atomic coherence / curved photonic lattice / electromagnetically induced transparency

1 Introduction

By substituting the time variable “t” in Schrödinger equation governing the motion of an electron with the coordinate “z”, there occurs an equation named as the paraxial wave equation, that describes the spatial propagation of light. This mathematical similarity between the models in two apparently different systems makes the quantum-optical analogy [1] possible. Such an analogy allows for the simulation of the extraordinary dynamics of electrons with rapid temporal evolutions in crystals by directly observing the spatial evolutions of a beam in media with modulated susceptibilities. In particular, the photonic crystals and lattices [2] with periodical susceptibilities have become promising analog platforms by guiding the propagation of light in desired manners. Up to now, considerable novel phenomena have been successfully revealed in optical systems, such as quantum tunneling [3, 4], Zeno dynamics [5, 6], Berry phase [7−9], and non-Hermitian properties [10−13], which were raised in quantum mechanics originally. In the meanwhile, these demonstrated results also bring out additional degrees of freedom for manipulating light.

Recently, curved photonic lattices, with the transverse bending of their central axis relating to the propagation direction, have attracted intensive interests. From the perspective of quantum-optical analogies, light travelling in curved photonic lattices could effectively mimic the behavior of electrons in artificial atomic crystals driven by an external field [14], which is associated with formation of the curved trajectory. As a result, fascinating concepts in solid-state physics, such as Bloch oscillation [15−19] induced by DC fields, Anderson localization [20−22] derived from disorder, and the dynamic localization [23−25] exerted by AC fields, have been proved in optical systems. Concretely, the introduction of AC field causes the transversely periodic motion of refractive index distribution with extending the longitudinal distance, while the DC field produces the spatial shift of refractive index distribution along only one transverse direction. As a consequence, the incident narrow wave packet expands over several channels and then returns to its initial state (namely dynamic localization), following the periodic change of the AC field, or oscillates with a frequency (namely Bloch oscillation) proportional to the amplitude of the DC field. Besides, the one-dimensional curved photonic lattice which mimics the effect of an external sinusoidal driving field has obtained a rich variety of achievements, such as dispersionless coupling [26], topological states [27−29], Floquet edge states [30], and Weyl interface states [31], to name a few. In general, such curved photonic lattice introduces an additional degree of freedom to control the propagation of light by breaking the parity symmetry [32], in contrast to the straight one.

Until now, the curved photonic lattices have predominantly been fabricated by laser direct writing [33] or interference/holographic lithography [34] in solid materials, and the light propagation inside the lattice is visualized by fluorescence imaging. Photonic lattices prepared in this way mostly possess fixed structures, and the modification of their parameters often necessitates the fabrication of new samples. The optically induced photonic lattice established in atomic vapors with the assistance of electromagnetically induced transparency (EIT) [35−40] has successfully overcome this limitation. Such electromagnetically induced photonic lattice can be regarded as electromagnetic induced grating (EIG) [41−43]. As a pioneering optical structure, EIG is characterized by Talbot self-imaging effect [44−46] which has been investigated in stationary straight structures constructed by fixed optical phase. Its spatial profile and band structures can be reconfigured by atomic coherence instantaneously [47, 48]. Recent researches including non-Hermitian delocalization [49], multiple vortex−antivortex pairs [50], angular-dependent Klein tunneling [51], and spin−orbit coupling [52] indicates that electromagnetically induced photonic lattices can be certainly regarded as a promising platform for quantum-optical analogies. The curved photonic lattices have also been proposed under the atomic coherent framework [16] by introducing frequency difference, but has not been experimentally verified yet.

In this paper, we put forward an approach to realize the sinusoidally curved photonic lattice in Rb atomic vapors by exploiting the phase difference in a two-beam interfering configuration. Compared to the previous works of EIG, an electro-optic modulator (EOM) is applied in the current work to sinusoidally modulating the phase of an optical beam, and gives rise to the sinusoidal translation (in the direction of the periodicity) of the interference fringes. With the assistance of EIT, the moving fringes establish a spatially distributed susceptibility, and eventually form a sinusoidally curved photonic lattice which resembles the introduction of the AC field. Theoretically, the calculated quasi-energy bands of the sinusoidally curved lattice collapse into a flat band when the modulation frequency meets specific values. In experiment, for clearly demonstrate the process of phase-difference-induced curved trajectory, we inject a Gaussian probe into the formed photonic lattice to observe the discrete diffraction [2] at a modulation frequency of 0.6 Hz. The output discrete pattern of the probe field moves sinusoidally over time, as the voltage signal applied to EOM does. This result advocates the establishment of the sinusoidally curved photonic lattice with controllable oscillating amplitude and speed. The current work provides a convenient and prospective platform for implementing quantum-optical analogies in curved photonic lattices.

2 Experimental scheme

The experimental setup is illustrated in Fig.1(a). The coupling field emitted from the external cavity diode laser is split into two beams E₂ and E₂′ (λ₂ ≈ 794.97 nm, vertical polarization) by using polarization beam splitters and half-wave plates. The two beams then intersect at the center of the atomic vapor cell with the angle of 2θ ≈ 0.2° to construct a standing-wave with a period of d ≈ 220 μm in the x direction. An EOM (Thorlabs EO-PM-NR-C1), whose refractive index can be linearly controlled by an applied voltage, is inserted into the propagation path of E₂′ to change its optical length in wavelength-level. When a sinusoidal high-voltage electrical signal is applied to the EOM, the phase difference between the two coupling beams also varies with the same sinusoidal frequency. As a result, the fringes formed by the two-beam interference also oscillate along the x axis. When the modulating speed is sufficiently high, the fringes propagating along the z axis exhibit a curved distribution in the x−z plane as shown in Fig.1(b). The amplitude, frequency or even shape of the oscillation are entirely determined by the electrical signal applied on the EOM. To some extent, this setting can also be regarded as an electrically controlled Mach−Zehnder interferometer [53].

A weak Gaussian probe field E₁ (λ₁ ≈ 794.98 nm, horizontal polarization) is also injected into the atomic vapor cell to form an EIT configuration together with the standing-wave coupling field. The involved Λ-type energy-level is shown in Fig.1(c), which consists of two hyperfine states F = 2 (level |1⟩) and F = 3 (level |2⟩) of ground state 5S_1/2 and one excited state 5P_1/2 (level |3⟩) of ⁸⁵Rb. The coupling field connects the transition |2⟩→|3⟩ and the probe field drives the transition |1⟩→|3⟩. The frequency detuning Δ_i (i = 1, 2) is defined as the difference between the laser frequency ω_i of field E_i and intrinsic resonant frequency of the corresponding atomic energy levels. Under the EIT condition, the susceptibility profile is “written” by the curved coupling field and can be analyzed with the help of the detected discrete probe patterns. The output of probe field is captured by a charge-coupled device (CCD) camera around the EIT window, whose center satisfies the two-photon resonant condition Δ₁ = Δ₂. The distance from CCD to L1, as well as the distance from L1 to the rear surface of the cell, is set to twice the focal length of L1, aiming for imaging the output pattern at its real size. Next to the rear surface of the vapor cell, a polarizer is placed to filter out the unwanted vertically polarized coupling fields. A portion of the output probe is directed onto the photodetector to monitor the EIT spectrum. The 5-cm-long atomic vapor cell is wrapped by μ-metal sheets and heat tapes to shield magnetic fields and adjust the atomic density, respectively.

3 Results and discussion

In theory, the spatial distribution of a stationary-state electromagnetic field can be described by complex amplitude function. In the presence of the coupling fields u₂(r) = A₂exp(ik₂r + φ₂) and u_2′(r) = A_2′exp(ik_2′r + φ_2′), the observed intensity at position r(x,y,z) can be derived as |u₂(r) + u_2′(r)|². In the current configuration, an extra optical phase of Δφ_2′ = Rcos(ω_tt) is synchronously introduced into u_2′(r) by the voltage signal applied on the EOM. Since the coupling field propagates along the z axis with a velocity v, the phase difference can be rewritten as Δφ_2′ = Rcos(ω_zz), where ω_z = ω_t/v. With the longitudinally dependent Δφ_2′, the spatial intensity distribution at the intersection of two beams also appears as a function of the propagating distance z inside the medium, and the curved period (in the z axis) is determined by the frequency of phase modulation. The linear refractive index in such an atomic system can be expressed as n = (1 + \chi)^1/2 ≈ 1 + \chi/2, where \chi is the linear susceptibility of the EIT medium. Considering n = n₀ + Δn, where n₀ = 1 is the background index of the atomic medium and Δn is the modulation of refractive index caused by the coupling field, one can obtain Δn ≈ \chi/2. The spatially modulated susceptibility in the Λ-type three-level ⁸⁵Rb atomic medium can be expressed as [54]

(1){\chi }^{(1)}={\displaystyle \frac{\mathrm{i}N{\left|{\mu }_{31}\right|}^2}{\hslash {\epsilon }_0}}\times {\left(({\mathrm{\Gamma }}_{31}+\mathrm{i}{\mathrm{\Delta }}_1)+{\displaystyle \frac{|{\mathrm{\Omega }}_2+{{\mathrm{\Omega }}^{\prime }}_2{|}^2}{{\mathrm{\Gamma }}_{32}+\mathrm{i}({\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2)}}\right)}^{-1},

where N is the atomic density; Γ_ij is the decay rate between states |i⟩ and |j⟩; Ω_g = µ_ijE_g/ħ is the Rabi frequency for the transition |i⟩ → |j⟩ with E_g being the electric field of the laser beam and µ_ij being the electric dipole momentum (i, j, and g being integer numbers). The coupling field Rabi frequency can be written as |{\mathrm{\Omega }}_2+{\mathrm{\Omega }}_2^{\prime }{|}^2=({\mu }_{12}/\hslash {)}^2[{E_2}^2+{E_2^{\prime }}^2+2E_2{E}_2^{\prime }\cos (2k_{c}x+\mathrm{\Delta } {\phi }_{2^{\prime }}) ], where k_c = (k₂ + k₂′)(sinθ)/2 with k₂ and k₂′ defined as the wave vectors of E₂ and E₂′, respectively. Considering the longitudinal modulation in the z direction, the centers of each waveguide move along x during its propagation with a period of Z, i.e., x_m(z) = x_m(z + Z), where m is the number of the corresponding waveguide.

The propagation dynamics of the probe field in such a susceptibility space are guided by the following Schrödinger-like paraxial equation [55]:

(2)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}\psi (x,z )=- {\displaystyle \frac{1}{2k_1}}{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}\psi ( x,z) -{\displaystyle \frac{k_1}{2}}{\chi }^{(1)}\psi (x ,z),

where ψ is the electric-field envelope of the incident probe beam; k₁ = 2πn₀/λ₁ is the wavenumber; and χ describes susceptibility distribution in the lattice consisting of waveguides with a transverse oscillating period of Z = 2π/ω_z and an amplitude of A = Rd/(2π). We therefore transform the coordinates into a reference frame where the waveguides are kept straight in the z direction, namely: x′ = x + x₀(z) with x₀(z) = Acos(ω_zz), and apply the gauge transformation \psi =\varphi \exp \{\mathrm{i}{k}_1[x^{\prime }(\mathrm{d}{x}_0/\mathrm{d}z)+\frac{1}{2} {\int }_0^z [\mathrm{d}{x}_0(\tau )/\mathrm{d}\tau {]}^2\mathrm{d}\tau ]\} [16], then Eq. (2) can be written as

(3)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z}}=-{\displaystyle \frac{1}{2k_1}}{\displaystyle \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{x^{\prime }}^2}}-{\displaystyle \frac{k_1}{2}}{\chi }^{(1)}\varphi -k_1{\ddot{x}}_0{x}^{\prime }\varphi .

Note that the bending effect of the central axis of the lattice is expressed in the last term in Eq. (3). By expanding the field into a superposition of the single-mode fields in individual waveguides as following:

(4)\varphi (x, z)={\displaystyle \sum _m{\phi }_m(z)s_m(x),}

one can obtain the coupled-mode equations of the waveguide array [30, 56]:

(5)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{\phi }_m}{\mathrm{\partial }z}}=c{\phi }_{m+1}+ c{\phi }_{m-1}-k_1{\ddot{x}}_{0}n \mathrm{d}{\phi }_m,

where φ_m is the amplitude in the mth waveguide, and c=k_1/2\int s_m^{\ast }(x)\chi (x,z)s_{m+1}(x)\mathrm{d}x is the coupling constant between the two neighboring waveguides. By performing a transformation {\phi }_m=\exp (\mathrm{i}{k}_1{\dot{x}}_{0}nd)a_m, we derive the coupled-mode equations as

(6)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }{a}_m}{\mathrm{\partial }z}}={\displaystyle \sum _{l=m\pm 1}c\exp (\mathrm{i}\gamma {\mathit{r}}_{lm})a_l,}

where γ(z) = −k₁Aω_zsin(ω_zz) is an effective gauge potential associated with the waveguide bending, and r_lm is the displacement between two neighboring waveguides l and m. Owing to the periodicity of coefficient γ(z) in Eq. (6), there is no static eigenmode. Instead, the solution to Eq. (6) is given by Floquet modes a_m(z) = u_m(z)exp(iβz), where u_m(z) evolves along z with the period Z, and the Floquet exponent β represents the quasi-energy band of the periodically curved waveguides.

Without loss of generality, we set the coupling constant c = 1 and calculate the susceptibility distributions according to Eq. (1) at given modulation frequencies ω_z of 0, 7 and 27 m⁻¹ as shown in Fig.2(a)−(c). One can see that with the increase of the modulation frequency, the expected sinusoidal motion of the refractive index occurs in z direction. The calculated band structures with modulating frequencies mentioned above are shown in Fig.2(d)−(f). Obviously, the gap between the upper and lower quasi-energy bands (marked as Δβ) is the largest for straight waveguides (ω_z = 0), as shown in Fig.2(d). When the modulating frequency increases, Δβ in curved waveguides significantly narrows down as shown in Fig.2(e) and eventually collapse into 0, indicating a flat band in Fig.2(f), which means that dynamic localization may occur [57]. The gap between the upper and lower quasi-energy bands versus ω_z is shown in Fig.2(g), and it is clear that the bands merge at some specific frequencies, such as ω_z = 27 m⁻¹. In fact, for this sinusoidal curved waveguide array with x₀(z) = Acos(ω_zz), the merging of bands is connected with the effective coupling coefficient c_eff = 0, where c_eff = cJ₀(k₁Aω_zd) and J₀ is the Bessel function of the first kind [58].

In fact, the maximum modulation frequency that can be achieved in our experiment is far lower than the frequency required for the formation of flat band. Hence, the real structure of the quasi-energy bands is closer to the band of the common straight waveguide array in Fig.2(a). But we can examine the dynamical evolutions of the probe field after passing through such a curved photonic lattice under a lower modulating frequency that reaches the limit of the adopted EOM. In this case, the simulated output probe at the rear surface of a practical 5-cm-long vapor cell versus time is shown in Fig.3(a) while the modulation angular frequency ω_t is 2π × 100 MHz. It is clear that the discretely diffracted probe field moves rapidly and periodically over time. Due to the extremely high propagation speed of the coupling field inside atomic vapors, the curving of the probe field is still very limited even in an assumed 80-cm-long vapor cell, as simulated in Fig.3(b). At the end surface of the vapor cell, the curved photonic lattice is cut off. Then the propagation of the discrete pattern in free space no longer has curved feature, obeying the standard free-space propagation and exhibiting the Talbot self-imaging effect [44−46] within a certain distance. Indeed, to experimentally achieve the flat band predicted by our model in a practical shorter atomic vapor, the modulating frequency may need to reach ~GHz.

To experimentally demonstrate the validity of the proposed mechanism vividly, a sinusoidal signal with a frequency of 0.6 Hz (ω_t = 2π × 0.6 Hz) is applied to the EOM to induce slow oscillation (along the x axis) of the one-dimensional photonic lattice. Here, the oscillating speed is set slow enough to allow one to clearly observe the dynamical displacement of the formed lattice via the CCD camera. We captured several sets of the output probe patterns at different times and plotted the displacement of the curved lattice over time in Fig.4. One can see that the discretely diffracted probe field presents a fringe-type distribution, revealing the spatial structure of the photonic lattice as expected. The orange dashed ellipses in Fig.4(a)−(g) mark the same fringe on the probe pattern, whose position varies with the phase difference. The stripe takes about 1.6 s to return to its original location marked by the white dotted line, indicating that the curved lattice does evolve over a whole period. The dynamical evolution of the transmitted probe patterns is provided as a video in Supplemental Material 1. Fig.4(h) shows the displacement at 0.1 s intervals (red circles) and plots the fitted result (white curve) assisted by a sinusoidal function. It is clear that the displacement of the photonic lattice along the x axis with time satisfies the sinusoidal function well, as the expected result of the sinusoidal voltage applied to EOM. The displacement frequency and amplitude (of 0.59 Hz and 212 μm) derived from the fitted expression agree with the applied signal and the constructed one-dimensional lattice constant, respectively. Increasing the frequency of the electrical signal can accelerate the speed of the curved lattice, and Supplemental Material 2 records the observations at a faster frequency of 1.2 Hz. Therefore, the proposed scheme has been proven to be feasible for producing curved photonic lattices, whose oscillation amplitude and velocity are instantaneously reconfigurable.

4 Conclusion

In conclusion, we use an electro-optic modulator to introduce an additional dimension of manipulation in one-dimensional photonic lattice constructed by two-beam interference. As the theoretical results turned out, the quasi-energy bands of the curved lattice collapses into a flat one at certain modulation frequency. The experimental results show that the diffracted probe field moves periodically with the rhythm of the signal applied to EOM, advocating the formation of the curved lattice. It is inspiring that the phase difference masters an excellent real-time control of lattice motion, including the intact trajectory and exact speed, and brings instantaneous reconfigurability to the curved photonic lattice in atomic vapors. However, limited by the modulation speed of the adopted free-space EOM (100 MHz in our work), the truly curved photonic lattice mentioned above that contains more than one cycle of oscillation (6-cm-long medium requires a modulation frequency of 5 GHz) is not realized in experiment. Consequently, the curved photonic lattice constructed through phase difference modulation may need a longer medium length compared to the conventionally actual waveguides fabricated with laser-direct writing technology, but possesses an obvious advantage of instantaneous tunability. The fiber-coupled EOM which has a maximum modulation frequency of 100 GHz would significantly help to reduce the medium length and achieve curved photonic lattices with several oscillating cycles. In addition to the sinusoidally curved photonic lattice proposed in current work, other potential structures with higher dimensions and more characteristics can also be constructed in this way. Although the currently accessible modulation speed is relatively slow, our work still opens the opportunity to explore quantum-optical analogies in optically induce reconfigurable photonic structures with both transverse and longitudinal modulation.

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