Coherent Control of Optical Nonreciprocal Propagation in a V-Type Atomic System

Optical nonreciprocity and nonreciprocal propagation of light have attracted great research interest, due to not only their fundamental scientific significance, but also their extensive applications in lasing, quantum optical devices and quantum information. In this work, we theoretically and experimentally investigate nonreciprocal propagation of light in a V-type three-level thermal atomic system. By virtue of the EIT effect and the atom thermal motion, nonreciprocal propagation of light is achieved in the Rb87 warm atoms, where high transmission of the probe field is achieved in the co-propagation direction of the control field and the probe field is blocked in the opposite direction of the control field. Transmission and bandwidth for the nonreciprocal propagation of light can be enhanced and controlled by the control field in this system, where the nonreciprocal band width can be broadened significantly in comparison with the Λ-type atomic system. In our experiments, we achieve ~60 MHz nonreciprocal bandwidth for the probe field. This work may have potential applications in quantum nonreciprocal devices such as optical isolator and circulator. on integration of devices. Great efforts dedicate to searching for alternative approaches and mechanisms to break reciprocity without the use of magnetism, especially those for suitable on-chip integration. A photonic band gap material with the combination of linear and nonlinear medium response previously proposed to support unidirectional propagation and optical diode [7]. Spatiotemporal modulation of refractive index of materials is one promising approach for this purpose, which generates optical nonreciprocity via introducing nonreciprocal phase transfer [8,9] and frequency conversion [10,11], Introduction Optical nonreciprocity and nonreciprocal devices, which supports drastically asymmetric propagation of light in two opposite directions, are essential in optical communications, laser systems and signal processing [1]. In recent years, optical nonreciprocity has attracted great research interest, and various strategies or physical mechanisms are proposed and studied for nonreciprocal transmission of light. Utilization of magneto-optic effect is a common approach to break the reciprocity [26]. However, response of magnetic materials often performs weak, implying bulky, costly and difficulty DOI: 10.35840/2631-5092/4523 • Page 2 of 9 • Qi et al. Int J Opt Photonic Eng 2020, 5:023 ISSN: 2631-5092 | Citation: Qi Y, Wang P, Niu Y, Gong S (2020) Coherent Control of Optical Nonreciprocal Propagation in a V-Type Atomic System. Int J Opt Photonic Eng 5:023 or establishing an angular momentum biasing [12-14]. Nonmagnetic optical nonreciprocity can also be achieved by optoacoustic effects [11,15], optical nonlinearity [16-19], and moving systems [20-22]. Great research interest were paid on the parity-time symmetry [23,24] recently. Using parity-time symmetric system, optical nonreciprocity [25] and phonon diode [26] have been studied. Due to the rapid development and the flexibility, optomechanical systems provide a good platform to support nonreciprocal transmission and create nonreciprocal devices such as optical isolator, optical circulator and optical router [27-34]. In chiral quantum physics, photons propagating in opposite directions are of spin-momentum or polarization locking, which drives emitters with different transition levels and rates [35]. It thus naturally offers a novel way to support nonreciprocal propagation of light even in the quantum regime [36-43]. The spin-orbit coupling canal so be used to realize optical nonreciprocity in low-dimensional materials [44-46]. Since the electromagnetically induced transparency (EIT) technology was introduced by Harris, et al. [47,48], many interesting quantum optical phenomena have been observed and realized in multi-level quantum systems based on the EIT effect, such as electromagnetically induced grating (EIG) [49-51], four-wave and six-wave mixing [52,53], optical bistability and multistability [54,55], optical switching [56], Kerr nonlinearity enhancement [57-60], weak-light optical solitons [61-66] etc. Via inducing periodic structures by lasers in the EIT atomic systems, dipole soliton and optical vortices were generated and studied experimentally in thermal atoms [67,68]. The random motion of atoms often takes disadvantage on quantum coherence in warm atoms. However, utilizing the atom thermal motion and EIT effect, our group experimentally investigated and achieved optical nonreciprocity and isolation in a cavity-atom coupling system [69]. And soon, a scheme of unidirectional amplification of light was also proposed and demonstrated in an atomic system [70]. Utilizing the optical nonlinearity of cross phase modulation, Xia, et al. theoretically proposed a scheme for optical isolator and optical circulator in an N-type thermal atomic system [71]. We also proposed a scheme to experimentally achieve optical nonreciprocity via optical pump effect in multi-level atomic systems [72]. Our recent experiment demonstrated that, the nonreciprocal bandwidth can be broadened in the cavity-free N-type atomic system [73]. Three level V-type atomic system is a very common and frequently used quantum system. In this work, based on the EIT effect, we experimentally and theoretically investigate the nonreciprocal propagation of light in a warm Rb87 V-type atomic system. By adjusting the control field, transmission and bandwidth can be controlled and enhanced for nonreciprocal propagation of the probe field in this system. It is shown that, high transmission of the probe light is achieved in the co-propagation direction of the control field, while it is effectively blocked in the opposite direction. In addition, the V-type atomic system can provide a relatively wider band width for nonreciprocal propagation of the probe light in comparison with the Λ-type atomic system. This work may provide reference for broadband applications of optical nonreciprocal devices. Model and Theoretical Analysis In this work, we consider a weak probe field of Rabi frequency p Ω and a strong control field of Rabi frequency c Ω interacting with a V-type atomic system (as shown in Figure 1). Under the slowly varying envelope and paraxial approximations, evolution of the probe field is governed by the following wave equation: p p p p 2 E k = i E z χ ∂ ∂ . (1) Transmission of the probe field is determined by the macro susceptibility p χ , which can be derived by solving the motion equations of the density matrix elements under steady states. Under electric-dipole and rotating-wave approximation, the interacting Hamiltonian can be written in the interaction picture as int c 22 p 33 c 21 p 31 ( ) ( c ) H = σ σ σ σ − ∆ + ∆ − Ω + Ω + Η. .   , (2) Where p ∆ and c ∆ indicate the one-photon detunings respectively for the probe and control lasers. They are defined as p p 31 = ω ω ∆ − and c c 21 = ω ω ∆ − with p ω and c ω being the angular frequencies of the probe and control lights and ( 31,21) ij ij = ω the relevant transition frequency between states i and j . 2 ( (p,c), (31,21)) l ij l = E l = ij = μ Ω ⋅ 


Introduction
Optical nonreciprocity and nonreciprocal devices, which supports drastically asymmetric propagation of light in two opposite directions, are essential in optical communications, laser systems and signal processing [1]. In recent years, optical nonreciprocity has attracted great research interest, and various strategies or physical mechanisms are proposed and studied for nonreciprocal transmission of light. Utilization of magneto-optic effect is a common approach to break the reciprocity [2-6]. However, response of magnetic materials often performs weak, implying bulky, costly and difficulty or establishing an angular momentum biasing [12][13][14]. Nonmagnetic optical nonreciprocity can also be achieved by optoacoustic effects [11,15], optical nonlinearity [16][17][18][19], and moving systems [20][21][22]. Great research interest were paid on the parity-time symmetry [23,24] recently. Using parity-time symmetric system, optical nonreciprocity [25] and phonon diode [26] have been studied. Due to the rapid development and the flexibility, optomechanical systems provide a good platform to support nonreciprocal transmission and create nonreciprocal devices such as optical isolator, optical circulator and optical router [27][28][29][30][31][32][33][34]. In chiral quantum physics, photons propagating in opposite directions are of spin-momentum or polarization locking, which drives emitters with different transition levels and rates [35]. It thus naturally offers a novel way to support nonreciprocal propagation of light even in the quantum regime [36][37][38][39][40][41][42][43]. The spin-orbit coupling canal so be used to realize optical nonreciprocity in low-dimensional materials [44][45][46].
Since the electromagnetically induced transparency (EIT) technology was introduced by Harris, et al. [47,48], many interesting quantum optical phenomena have been observed and realized in multi-level quantum systems based on the EIT effect, such as electromagnetically induced grating (EIG) [ 66] etc. Via inducing periodic structures by lasers in the EIT atomic systems, dipole soliton and optical vortices were generated and studied experimentally in thermal atoms [67,68]. The random motion of atoms often takes disadvantage on quantum coherence in warm atoms. However, utilizing the atom thermal motion and EIT effect, our group experimentally investigated and achieved optical nonreciprocity and isolation in a cavity-atom coupling system [69]. And soon, a scheme of unidirectional amplification of light was also proposed and demonstrated in an atomic system [70]. Utilizing the optical nonlinearity of cross phase modulation, Xia, et al. theoretically proposed a scheme for optical isolator and optical circulator in an N-type thermal atomic system [71]. We also proposed a scheme to experimentally achieve optical nonreciprocity via optical pump effect in multi-level atomic systems [72]. Our recent experiment demon-strated that, the nonreciprocal bandwidth can be broadened in the cavity-free N-type atomic system [73]. Three level V-type atomic system is a very common and frequently used quantum system. In this work, based on the EIT effect, we experimentally and theoretically investigate the nonreciprocal propagation of light in a warm Rb87 V-type atomic system. By adjusting the control field, transmission and bandwidth can be controlled and enhanced for nonreciprocal propagation of the probe field in this system. It is shown that, high transmission of the probe light is achieved in the co-propagation direction of the control field, while it is effectively blocked in the opposite direction. In addition, the V-type atomic system can provide a relatively wider band width for nonreciprocal propagation of the probe light in comparison with the Λ-type atomic system. This work may provide reference for broadband applications of optical nonreciprocal devices.

Model and Theoretical Analysis
In this work, we consider a weak probe field of Rabi frequency p Ω and a strong control field of Rabi frequency c Ω interacting with a V-type atomic system (as shown in Figure 1). Under the slowly varying envelope and paraxial approximations, evolution of the probe field is governed by the following wave equation: Transmission of the probe field is determined by the macro susceptibility p χ , which can be derived by solving the motion equations of the density matrix elements under steady states. Under electric-dipole and rotating-wave approximation, the interacting Hamiltonian can be written in the interaction picture as Where p ∆ and c ∆ indicate the one-photon detunings respectively for the probe and control lasers. fields.
The macro polarization for the probe field is p 0 p p 13 31 ε is the vacuum permittivity and 31 ρ represents the corresponding density matrix element. As we consider in the warm atomic gas, the frequencies of the lasers felt by the atoms depend on not only the frequencies of the incident lasers but also the frequency shift caused by the atom Γ indicates the decoherence and population decay rates respectively. It can be seen in Figure 2 that, the transmission line widths V W and L W of the forward probe field increases with the Rabi frequency of the control field c Ω (or the intensity of the control field), and the transmission line width V W can be much broader than L W especially for small c Ω . V W can be dozens or even hundreds of times larger than L W under the same c Ω .

Experiment for Nonreciprocal Propagation of Light
We experimentally investigate the nonreciprocal propagation of light in a warm V-type Rb87 atomic system. We select the levels (5 2 S 1/2 , F = 2), (5 2 P 1/2 , F = 2) and (5 2 P 3/2 , F = 3) of Rb87 atoms as the states 1 , 2 and 3 and set laser couplings as shown in Figure 1a. The strong control laser c Ω with wavelength of 780 nm is applied to drive the transition 1 2 ↔ . A weak laser p Ω with wavelength of 795 nm is used to probe the 1 3 ↔ transition. Such consideration of laser excitation forms a V-type configuration. The experimental setup and laser paths in experiment are laid out as shown in Figure 1b. With such arrangement of light paths, the coupling lasers c Ω is vertically polarized and the probe laser is the effective detuning with the wave vector of the lasers k  and the atom velocity v  .
k T M is the most probable velocity with the Boltzmann constant B k , the absolute temperature T , and the atom mass M . The superscript (F, B) indicates the forward or backward propagation direction for short. Similarly, we can also obtain the susceptibility for the Λ-type atomic system. Transmission of the probe field in the two atomic systems can be calculated by Eqs.
When the control field propagates along the forward direction, the atomic thermal motion gives rise to the same frequency shift on the forward probe field and opposite frequency shift on the backward probe field if So transmission of the backward probe field can be greatly suppressed due to the destruction of EIT, while transmission of the forward probe field can maintain a high level. Then nonreciprocal propagation of the probe field can be achieved in the two opposite directions. The nonreciprocal bandwidth is mainly determined by the EIT line width of the forward probe field and the Doppler line width of the warm atoms. Figure  2 shows the comparison of the transmission line width of the forward probe field in the V and Λ-type warm atomic systems by solving the Eqs.  Figure 3c and Figure 3d respectively. The probe field obviously realizes high transmission in the forward direction because of the EIT effect (Figure 3c). While in the backward direction, transmission of probe field is very low under certain control powers (Figure 3d). Therefore, nonreciprocal propagation of probe field can be achieved in the two opposite directions.
We further measure bandwidth of the nonreciprocal propagation by evaluating the full width at half maximum (FWHM) of the forward transmission in the EIT window and show the results in . For convenience, we define the path from left to right as the forward direction whereas the path from right to left as the backward direction. In our experiment, propagation direction of the control field is fixed to be along the forward direction. Figure 3a and Figure 3b show the transmissions of the forward and backward probe fields versus the probe detuning and laser power P c of the control field respectively, where the average background noise has been erased. As shown in Figure 3a, we obtain high transmission for the forward probe field near the resonant frequency. The transmission and bandwidth depend sensitively on the power of the control field P c . It is obvious that, with the increase of P c , transmission and band width of the probe field are significantly enhanced (see Figure 3a). Contrarily, as the thermal motion of atoms induces opposite frequency shift for the backward probe field and breaks the EIT effect, backward transmis-

Conclusion
In conclusion, based on the electromagnetically induced transparency and the thermal motion of the atoms, we have investigated nonreciprocal propagation of light in a V-type Rb87 warm atomic system. Transmission of the probe field in two opposite directions can be controlled and enhanced by adjusting the control field. In the same propagation direction of the control field, the thermal motion of the atoms causes approximate frequency shift on the probe field, which guarantees the EIT condition and produce high transmission of the probe field. Conversely, in the opposite direction, the thermal motion induced frequency shift is nearly contrary between the probe field and the control field. So the probe field can be significantly blocked in the backward direction. In addition, bandwidth 19. Y Shi, Z Yu, S Fan (2015) Limitation of nonlinear optical isolators due to dynamic reciprocity. Nat Photon 9: 388-392.