• Chinese Optics Letters
  • Vol. 20, Issue 1, 012701 (2022)
Shengfa Fan1、2, Yihong Qi1、*, Yueping Niu1, and Shangqing Gong1
Author Affiliations
  • 1School of Physics, East China University of Science and Technology, Shanghai 200237, China
  • 2School of Materials Science and Engineering, East China University of Science and Technology, Shanghai 200237, China
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    DOI: 10.3788/COL202220.012701 Cite this Article Set citation alerts
    Shengfa Fan, Yihong Qi, Yueping Niu, Shangqing Gong. Nonreciprocal transmission of multi-band optical signals in thermal atomic systems[J]. Chinese Optics Letters, 2022, 20(1): 012701 Copy Citation Text show less

    Abstract

    Multi-band signal propagation and processing play an important role in quantum communications and quantum computing. In recent years, optical nonreciprocal devices such as an optical isolator and circulator are proposed via various configurations of atoms, metamaterials, nonlinear waveguides, etc. In this work, we investigate all-optical controlled nonreciprocity of multi-band optical signals in thermal atomic systems. Via introducing multiple strong coupling fields, nonreciprocal propagation of the probe field can happen at some separated frequency bands, which results from combination of the electromagnetically induced transparency (EIT) effect and atomic thermal motion. In the proposed configuration, the frequency shift resulting from atomic thermal motion takes converse effect on the probe field in the two opposite directions. In this way, the probe field can propagate almost transparently within some frequency bands of EIT windows in the opposite direction of the coupling fields. However, it is well blocked within the considered frequency region in the same direction of the coupling fields because of destruction of the EIT. Such selectable optical nonreciprocity and isolation for discrete signals may be greatly useful in controlling signal transmission and realizing selective optical isolation functions.

    1. Introduction

    Similar to their electronic counterpart, optical nonreciprocal devices play an important role and possess the fundamental function in photonic devices and quantum circuits, urging great research interest in an optical isolator[13], circulator[46], and router[7,8]. These optical nonreciprocal devices allow photon transportation in one direction, while blocking it in the reverse direction. In laser systems, the optical isolator, which blocks the back transmission of light, provides effective protection for the lasers. Optical nonreciprocal devices also have promising applications in all-optical quantum networking and computing. Transmission and processing of multi-wavelength/band optical signals is an important issue in optical communications and processing. Optical devices for multi-wavelength applications have attracted intensive attention in recent years, such as multi-wavelength lasers[913], multiplexing and communications[14,15], imaging and sensors[1619], and photovoltaic devices[20]. It is also a very important subject to develop and design optical nonreciprocal devices such as optical isolators suitable for multi-wavelength applications[21,22]. In this work, we are committed to the study of controllable multi-band optical nonreciprocal transmission by using atomic systems.

    The key to realize optical nonreciprocal devices is the nonreciprocal or asymmetric transmission of light. Magneto-optical materials were used to produce optical nonreciprocal propagation via the Faraday rotation effect[2327]. However, responses of the magnetic materials are often very weak, implying requirements of the large size bulk magnetic media or the strong magnetic fields, which may bring about some unfavorable impacts[1]. Then, for avoiding the use of magnetic materials, schemes based on optical nonlinearity[2831] or photonic crystal heterojunctions[32] were proposed. Recently, a number of works dedicated to realizing nonmagnetic optical nonreciprocal transmission via different schemes, such as frequency conversion[33,34], angular momentum biasing[3537], optoacoustic effects[34,38], artificial gauge field[3941], parity-time-symmetry breaking[4245], and moving medium[46,47].

    Generally, random thermal motion of atoms has a negative impact on coherence of the quantum system, resulting in the decoherence effect or thermal noise[4850]. It is fortunate that via a smart design people can also actively utilize the atomic thermal motion in some particular fields. Based on the electromagnetically induced transparency (EIT) effect and the thermal motion of atoms, Zhang et al. proposed a novel mechanism to achieve optical propagation of the probe field in a three-level Λ-type atom-cavity coupling system[51]. Then, Xia et al. investigated the direction-dependent cross phase modulation (XPM) in an N-type thermal atomic system and utilized the XPM to achieve an optical isolator and circulator[52]. Later, Gong et al. also investigated directional optical amplification[53], optical isolation by the pumping effect[54], and broadband optical nonreciprocity[55] in multi-level atomic systems. Manipulation of multiple optical signals may have potential applications in optical communications and quantum information processing. Utilizing dynamically induced photonic band gaps, Yang et al. have proposed a scheme to generate two-color optical nonreciprocity in a cold tripod-type atomic system[56]. In this work, stimulated by these works, we investigated all-optical controlled nonreciprocal propagation of multi-band optical signals in a Y-type-like multi-level hot atomic system. By introducing multiple strong coupling fields, some nonreciprocal bands with separated frequencies can be generated for the weak probe field. Moreover, these separated nonreciprocal frequency bands can be flexibly controlled by the coupling fields. This work may have potential applications in multi-band signal detection, discrimination, and processing.

    2. Model and Equations

    We consider interaction of laser fields and the atomic system of N+2 levels, as shown in Fig. 1, in which the weak probe field and N strong coupling fields couple corresponding energy levels in Fig. 1(a) and propagate along different directions, respectively, as shown in Figs. 1(b) and 1(c). The weak probe field Ωp of frequency ωp couples states |g and |m, and the transitions |m|n(n=1,…,N) are driven by the strong coupling fields Ωn of frequency ωn, where Ωp=|μgm|Ep/2 and Ωn=|μmn|En/2 are corresponding half-Rabi frequencies of the fields with the electric dipole momentum μij(i,j=g,m,1,…,N) and the electric field amplitudes Ep and En. The atomic gas is loaded in a cell, and its temperature is controlled by a temperature control system. In general, the atoms are in constant thermal motion following the Maxwell velocity distribution. Under the electric dipole and rotating-wave approximations, the Hamiltonian of the system can be written in the interaction picture as Hint=[Δp|mm|+n=1N(Δp+Δn)|nn|+(Ωp|mg|+n=1NΩn|nm|+H.C.)],where Δp=ω¯mgωp(Δn=ω¯nmωn) denotes the detuning of the probe field (coupling fields) for the corresponding transition |g|m(|m|n) with transition frequency ω¯mg(ω¯nm). From the Liouville equation, we can obtain the following motion equations for the density-matrix elements: ρ˙gg(t)=iΩpρmg(t)iΩp*ρgm(t)+n=1NΓngρnn(t)+Γmgρmm(t),ρ˙mm(t)=iΩpρgm(t)+in=1N[Ωnρnm(t)Ωn*ρnm(t)]iΩp*ρmg(t)Γmgρmm(t)+n=1NΓnmρnn(t),ρ˙nn(t)=iΩn*ρmn(t)iΩnρnm(t)Γnmρnn(t)Γngρnn(t),ρ˙mg(t)=(iΔpγmg)ρmg(t)+in=1NΩnρng(t)+iΩp*ρgg(t)iΩp*ρmm(t),ρ˙ng(t)=(iΔp+iΔnγng)ρng(t)+iΩn*ρmg(t)iΩp*ρnm(t),ρ˙nm(t)=i(Δn+iγnm)ρnm(t)+iΩn*ρmm(t)iΩpρng(t)il=1NΩl*ρnl(t),ρ˙nl(t)=i(ΔnΔl+iγnl)ρnl(t)+iΩnρml(t)iΩl*ρnm(t),nl,with n=1Nρnn(t)+ρgg(t)+ρmm(t)=1, ρij(t)=ρji*(t)(i,j=g,m,n;ij), and n,l=1N. We denote the radiative decay rate of the populations from level |n to |m (|g) by Γnm (Γng) and the decoherence rate by γnl, respectively. Assuming Ωp<Ωn,Γij, the atoms are mainly populated on state |g. Then, by solving the density matrix of Eqs. (2)–(8) in steady state, we can obtain the linear susceptibility for the weak probe field as χND|μgm|2Δp+iγmgn=1NΩn2/(Δp+Δn+iγng),with the atom density ND.

    Interaction of the laser fields and the multi-level atomic systems. (a) Laser coupling scheme, (b) co-propagation, and (c) counter-propagation setups of the probe field and the strong coupling fields.

    Figure 1.Interaction of the laser fields and the multi-level atomic systems. (a) Laser coupling scheme, (b) co-propagation, and (c) counter-propagation setups of the probe field and the strong coupling fields.

    Due to the irregular thermal motion, atoms in the hot atomic system move with various velocities in different directions. Under this condition, both frequencies of the lasers and frequency shift arising from the Doppler effect take effect on the interaction between lasers and atoms. Then, the detuning Δi(i=p,n) in Eq. (9) should be rewritten as Δi±kivj with the atom of velocity vj and the wavevector ki of the laser Ωi. Assuming all the coupling lasers Ωn propagate along the same direction, the effective macro susceptibility for the probe field Ωp of co-/counter-propagation with the coupling lasers should be integrated on all the atoms of different velocities by χco(cou)=+χD(v)dv,in which D(v)=ev2vp2/(πvp) indicates the Maxwell–Boltzmann distribution of the atoms, and vp=2kBT/M is the most probable velocity with the Boltzmann constant kB, the absolute temperature T, and the atom mass M. Then, susceptibilities for the co-/counter-propagating probe field can be obtained via the following integrations: χco=+ND|μgm|2D(v)Δpkv+iγmgS1dv,χcou=+ND|μgm|2D(v)Δp+kv+iγmgS2dv,with S1=n=1NΩn2/(Δp+Δn2kv+iγng),S2=n=1NΩn2/(Δp+Δn+iγng),where we have assumed kpki=k(i=1,…,N) for simplicity. Transmission of the probe field in the medium can be obtained from the Maxwell equations as follows: tco(cou)=eiαχco(cou)L,with the transmission coefficient α=ωp2ε0c and medium length L. Then, the normalized transmissivity for the probe field is Tco(cou)=|tco(cou)|2.

    So, transmission of the probe field in the co-/counter-propagation direction can be regulated to pursue high asymmetric transmission in the two opposite directions by controlling the coupling fields.

    3. Results and Discussion

    Figure 2 shows transmissions of the co-propagating (red dash-dotted line) and the counter-propagating (blue solid line) probe fields in the three, four, five, and six-level atomic systems. In the calculation, we consider the atomic medium length L=5.0cm, the atomic density ND=5.0×1016m3, the temperature T=70.0°C, and the other parameters are normalized by γ=γmg. It is clear that multiple nonreciprocal windows with separated frequency bands are generated in the multi-level atomic systems. When the probe field propagates along the opposite direction of the coupling fields, effects of the atom motion on the probe field and the coupling fields can be offset in the proposed configuration, which leads to the construction of the EIT under two-photon resonance (Δp+Δn=0) and thus high transmissivity of the probe field in the EIT windows. Then, it can be seen that one or several separated high transmission bands (blue solid lines) are created in the transmission spectrum for the probe field, depending on the number and detuning of control fields. However, for the co-propagating probe field, frequency shifts induced by the atom motion produce remarkable two-photon detunings for the probe field and the coupling fields, which destruct the EIT effect and make large absorption of the probe field. Under this condition, the weak probe field interacts with an effective two-level atomic system, and probe photons are greatly absorbed by a large number of atoms. The following spontaneous emission can never generate a field along the incident direction of the probe field. Transmission of the co-propagating probe field is almost vanishing (red dashed-dotted lines) in a wide spectrum range. Therefore, multi-band nonreciprocal propagation of the probe field can be achieved in this atomic system by introducing multiple coupling lasers driving corresponding transitions.

    Transmission of the probe field in multi-level atomic systems as a function of the probe detuning Δp, where the blue solid line stands for the counter-propagation and the red dash-dotted line for co-propagation. (a) Three-level system with Δ1 = 0, Ω1 = 40γ; (b) four-level system with Δ1 = −20γ, Δ2 = 20γ, Ω1 = Ω2 = 40γ; (c) five-level system with Δ1 = − 20γ, Δ2 = 0, Δ3 = 20γ, Ω1 = Ω2 = Ω3 = 40γ; and (d) six-level system with Δ1 = − 20γ, Δ2 = −10γ, Δ3 = 10γ, Δ4 = 20γ, Ω1 = Ω2 = Ω3 = Ω4 = 40γ.

    Figure 2.Transmission of the probe field in multi-level atomic systems as a function of the probe detuning Δp, where the blue solid line stands for the counter-propagation and the red dash-dotted line for co-propagation. (a) Three-level system with Δ1 = 0, Ω1 = 40γ; (b) four-level system with Δ1 = −20γ, Δ2 = 20γ, Ω1 = Ω2 = 40γ; (c) five-level system with Δ1 = − 20γ, Δ2 = 0, Δ3 = 20γ, Ω1 = Ω2 = Ω3 = 40γ; and (d) six-level system with Δ1 = − 20γ, Δ2 = −10γ, Δ3 = 10γ, Δ4 = 20γ, Ω1 = Ω2 = Ω3 = Ω4 = 40γ.

    In this scheme, each band for nonreciprocal propagation of light can be well controlled and shifted individually by changing the detuning of the corresponding coupling field, which brings us great convenience for optical signal or information processing. Figure 3 shows the transmissions of the co-propagating and counter-propagating probe fields with different detunings of the coupling fields in the five-level atomic system. It can be seen in Figs. 3(a)3(c), with the fixed detunings Δ1 and Δ3, the central frequency of the middle nonreciprocal band is shifted independently by tuning Δ2. Clearly, the left and right nonreciprocal bands can also be controlled by changing Δ1 and Δ3, respectively. Such frequency-tunable multi-band nonreciprocity may be very helpful in the processing of optical multi-band signals.

    Tunable nonreciprocal frequency bands in the five-level atomic system with Δ2 = 0, Ω1 = Ω2 = Ω3 = 40γ, Δ1 = −30γ, Δ3 = 30γ and (a) Δ2 = 20γ; (b) Δ2 = 0γ; (c) Δ2 = −20γ. The other parameters are the same as in Fig. 2.

    Figure 3.Tunable nonreciprocal frequency bands in the five-level atomic system with Δ2 = 0, Ω1 = Ω2 = Ω3 = 40γ, Δ1 = −30γ, Δ3 = 30γ and (a) Δ2 = 20γ; (b) Δ2 = 0γ; (c) Δ2 = −20γ. The other parameters are the same as in Fig. 2.

    Bandwidth of optical nonreciprocal devices plays an important role in applications[57,58]. Broad and tunable widths of nonreciprocal windows are often desirable for nonreciprocal devices such as an optical isolator and circulator. In this scheme, these nonreciprocal windows can be controlled individually or simultaneously by adjusting intensities of the coupling fields. To further examine the dependence of transmissivity and nonreciprocal bandwidth of the probe field on Rabi frequencies of the coupling fields, we calculated transmissions of co-propagating and counter-propagating probe fields in the five-level atomic system by changing the Rabi frequencies of the coupling fields. In Figs. 4(a) and 4(b), for simplicity, we adjust simultaneously the Rabi frequencies of the coupling fields by Ω1=Ω2=Ω3=Ω0. It is shown that, for the counter-propagating probe field, bandwidths of the central three separated high transmission bands increase simultaneously with the Rabi frequencies of the coupling fields, while transmission of the co-propagating probe field is well suppressed in corresponding frequency windows, and the total width of the absorption window is also broadened with the Rabi frequency of the coupling fields. In Figs. 4(c) and 4(d), we fix Ω1=40γ and Ω3=20γ and only change Ω2 to control transmission of the probe field. In this case, only the bandwidth of the central nonreciprocal window is enlarged with the increase of Ω2, but the left and right ones are almost unchanged.

    Variation of transmission of probe fields with (a), (b) Δp and Ω0 or (c), (d) Ω2, where (a), (c) are the results for the counter-propagating probe field and (b), (d) for the co-propagating probe field. In the calculation, Δ1 = −30γ, Δ2 = 0, Δ3 = 30γ, and the other parameters are the same as in Fig. 2.

    Figure 4.Variation of transmission of probe fields with (a), (b) Δp and Ω0 or (c), (d) Ω2, where (a), (c) are the results for the counter-propagating probe field and (b), (d) for the co-propagating probe field. In the calculation, Δ1 = −30γ, Δ2 = 0, Δ3 = 30γ, and the other parameters are the same as in Fig. 2.

    To further examine transmissivity and contrast η=|TcouTcoTcou+Tco| for the case of Figs. 3(c) and 3(d), we calculate and plot the transmissivity and corresponding transmission contrast at the central frequencies of the three nonreciprocal bands in Fig. 5. As shown in Fig. 5(a), transmission of the probe field at Δp=0 is enhanced obviously with the increase of Ω2, while at the other two nonreciprocal bands the probe fields have little change. This provides us with a way to flexibly control transmission of signals in need in the nonreciprocal windows. It is anticipated that high transmissions of the probe fields at different nonreciprocal bands can be achieved via increasing the corresponding intensities of the coupling fields. Figure 5(b) shows high transmission contrasts at the center of the three nonreciprocal bands, implying excellent isolation performance of them.

    Variations of (a) transmissivity T of the counter-propagating probe field and (b) corresponding transmission contrast η with Rabi frequency of the coupling field Ω2 at the center frequencies of the three nonreciprocal bands, corresponding to the cases of Figs. 4(c) and 4(d).

    Figure 5.Variations of (a) transmissivity T of the counter-propagating probe field and (b) corresponding transmission contrast η with Rabi frequency of the coupling field Ω2 at the center frequencies of the three nonreciprocal bands, corresponding to the cases of Figs. 4(c) and 4(d).

    It is necessary to have further discussion on the experimental feasibility and possible atomic systems. In the calculation, we have assumed kpki for simplicity, where the effect for the Doppler shift can be well canceled for the probe field in the counter-propagating directions. Generally speaking, it is not easy to find proper atomic transitions. However, Zeeman splitting levels in an alkali metal atomic system such as rubidium and cesium can provide a feasible way for realizing this model. For example, we can choose the transition 5S1/2,F=15P3/2,F=2 (384.23034 THz) as the probe field coupled levels and make the control fields with different polarizations couple the transitions 5P3/2,F=2(mF=0)5D3/2,F=1(mF=±1,0) (386.25231 THz) in Rb-87 atoms [as shown in Fig. 6(a)]. Then, in similar configurations, it can be greatly guaranteed that kpki. Even for kpki, the absorption can still be largely reduced in the counter-propagating direction, and thus nonreciprocity forms due to the partially eliminated Doppler effect and Doppler broadened linewidth of thermal atoms. For example, we can choose the atomic system and the laser coupling scheme, as shown in Fig. 6(b), where the probe laser couples the transition 5S1/2,F=15P3/2,F=2, and the control laser couples the transitions 5P3/2,F=25D3/2,F=1 (5D5/2,F=1; 7S1/2,F=1; 8S1/2,F=1). In this case, ωp=2π×384.23034THz and ω1,2,3,4=2π×(386.25231,386.3411,404.5667,486.58499)THz are used for calculation. It can be found that, as long as kp and ki are not too different, the property of nonreciprocity can be well kept. The only difference is that the transmission of the probe field in the counter-propagating direction may be suppressed slightly, or part of the multi-band signals cannot be well separated (as shown in Fig. 7). Therefore, multi-band nonreciprocity can also be achieved by using similar multi-level transitions in alkali-metal atoms, such as rubidium and cesium. For example, transitions of (5S1/2)(5P1/2,5P3/2)(6S1/2,7S1/2,8S1/2,…,5D3/2,6D3/2,7D3/2,8D3/2) in rubidium provide the possibility of cascade-like transitions. In addition, small tilt angles between the probe and coupling fields may also be arranged for matching the condition of kpki in experiment.

    Possible atomic systems and laser coupling schemes in experiments, where probe and control fields are with (a) adjacent frequencies by using Zeeman splitting levels and (b) different frequencies.

    Figure 6.Possible atomic systems and laser coupling schemes in experiments, where probe and control fields are with (a) adjacent frequencies by using Zeeman splitting levels and (b) different frequencies.

    Transmission of the probe field in counter-propagating (blue solid line) and co-propagating (red dashed line) directions by using the scheme in Fig. 6(b), where Δ1 = −40γ, Δ2 = −10γ, Δ3 = 10γ, Δ4 = 40γ, and other parameters are the same as in Fig. 2(d).

    Figure 7.Transmission of the probe field in counter-propagating (blue solid line) and co-propagating (red dashed line) directions by using the scheme in Fig. 6(b), where Δ1 = −40γ, Δ2 = −10γ, Δ3 = 10γ, Δ4 = 40γ, and other parameters are the same as in Fig. 2(d).

    4. Conclusions

    In conclusion, based on the EIT effect, we have investigated controllable multi-band nonreciprocal propagation of optical signals in the thermal multi-level cascade atomic systems. By use of multiple strong coupling fields, the weak probe field can propagate with several separated high transmission bands in the opposite direction of the coupling fields due to the EIT effect, while the co-propagating probe field can be well absorbed in the same frequency domain. This provides the possibility of generating and flexibly controlling multi-band nonreciprocal propagation of optical signals. Moreover, separation, bandwidth, and center frequencies of these nonreciprocal transmission bands can be well adjusted and controlled by changing the Rabi frequencies and detunings of the coupling lasers. Simultaneously, high transmission contrast can be maintained in these nonreciprocal bands, guaranteeing excellent optical isolation performance. This work may provide references for related optical isolation devices such as an optical diode and circulator. Other probable functions of the separated nonreciprocal bands may be extracting and discriminating optical signals, which may find application in optical information processing and optical networking.

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    Shengfa Fan, Yihong Qi, Yueping Niu, Shangqing Gong. Nonreciprocal transmission of multi-band optical signals in thermal atomic systems[J]. Chinese Optics Letters, 2022, 20(1): 012701
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