The manipulated left-handedness in a rare-earth-ion-doped optical fiber by the incoherent pumping field
aa r X i v : . [ phy s i c s . op ti c s ] D ec The manipulated left-handedness in a rare-earth-ion-dopedoptical fiber by the incoherent pumping field
Shun-Cai Zhao, ∗ Hong-Wei Guo, and Xiao-Jing Wei
Department of Physics, Faculty of Science, Kunming University of Science and Technology, Kunming, 650500, PR China (Dated: December 20, 2018)The left-handedness was demonstrated in an E r -dopped Z r F - B a F - L a F - AlF - N a F (ZBLAFN)optical fiber modeled by a three-level quantum system. Under the electric and magnetic componentsof the probe field driving the transitions of I / - I / and I / − I / in the E r -doppedoptical fiber respectively, an increasing left-handedness was achieved by the incremental incoherentpumping field. However, the left-handedness damped when the incoherent pumping field drovethe transition heavily. Our scheme may provide a solid candidate other than the coherent atomicvapour for left-handedness, and may extend the application of the rare-earth-ion-doped optical fiberin metamaterials via the external incoherent pumping field. Keywords: Rare-earth-ion-doped optical fiber; three-level quantum system; left-handedness; incoherentpumping field
I. INTRODUCTION
Optical fibers doped with rare-earth ions, such as E r , N d , T m , S M , H o , Y b , and P r , have attractedsignificant scientific and industrial interests due to theirapplications in optical fiber amplifiers and fiber lasers[1–3]. Especially, the E r -doped optical fiber plays a muchmore important role in optical fiber communication[4–11]. And the emission transition I / − I / in E r -doped fiber amplifier is the key element of mod-ern telecommunication systems[4, 5]. In wavelength di-vision multiplexing[6], the I / − I / emission andthe I / - I / absorption transitions of E r fulfill flatemission spectra and wavelength divergence. The transi-tions of H / − I / and F / − I / under infraredradiation excitation can convert fluorescence propertiesvia E r -doped nanoparticles of GdP O [12].On account of the flexible design and adjustable pa-rameters comparing to their atomic counterparts, the E r -doped optical fiber systems play a more importantrole in practical application for quantum optics. Andsome nonlinear quantum optical phenomena, such as gainleveling[13], optical bistability and multi-stability[14],absorption-amplification response[15], enhanced index ofrefraction with vanishing absorption[16] were achieved in ∗ Corresponding author: [email protected] E r -doped optical fibers recently.In this work, under the electric and magnetic compo-nents of the probe field coupling the transitions of I / - I / and I / − I / , respectively, we theoreticallyinvestigate the feasibility of left-handedness via the inco-herence pumping field in the E r -dopped ZBLAFN op-tical fiber. For the simplify of experimental realization,we simulate the E r -dopped optical fiber with an ordi-nary three-level quantum system, and the switching fromincreasing to decreasing left-handedness can be imple-mented simply by adjusting the pumping rate of the in-coherent pumping field. In our scheme, a new left-handedmaterial may be explored instead of artificial compositemetamaterials[17], photonic crystal structures[18], trans-mission line simulation[19] and chiral media [20], whichmay extend the mediums for left-handedness and the ap-plication domain for E r -dopped optical fiber in meta-materials II. MODEL AND EQUATION
The E r -dopped ZBLAFN optical fiber can be mod-eled by three-level quantum system shown in Fig. 1.The levels | I / i , | I / i and | I / i in the E r willbehaved the | i , | i and | i states, respectively. In sucha modeled 3-level system, the parities of levels | i and | i are set to be identical, which is opposite to | i . An in-coherent pump field pumps atoms in level | i into upperlevel | i with its pumping rate being 2Γ. The possibleoptical transition | i ↔ | i is mediated by a weak probelaser field with central frequency ω e and Rabi frequencyΩ e = ~E e d / ~ . Because of the parity selection rules, thetwo levels | i and | i with electric dipole element d = h | ˆ ~d | i6 = 0 are coupled by the electric component ofthe weak probe field, where ˆ ~d is the electric dipole op-erator. The magnetic component of the probe field withfrequency ω b and Larmor frequency of Ω b = ~Bµ / ~ isapplied to the magnetic-dipole transition | i↔| i , where µ is the corresponding magnetic-dipole matrix element.Interestingly, the transitions | i ↔ | i , | i ↔ | i and | i↔ | i coupled by the incoherent pump field, the electricand magnetic components of the probe laser field respec-tively form a closed-loop configuration. FIG. 1. Schematic diagram E r -dopped Z r F - B a F - L a F - AlF - N a F (ZBLAFN) in optical fibers modeled by a three-level system. The levels I / , I / , and I / behave the | i , | i and | i state labels, respectively Then the semi-classical interaction Hamiltonian H int of this ionic system in an E r -dopped ZBLAFN opticalfiber is given in the interaction picture under the dipoleand the rotating wave approximation as follows, H int = ∆ e ( | ih | + | ih | ) + ∆ b | ih | + (Ω b | ih | + Ω e | ih | + H.c. ) (1)where H.c. means Hermitian conjugation, ∆ e = ω e - ω and ∆ b = ω b - ω are the detunings of the electric andmagnetic components of the probe laser field to the tran-sitions ω and ω , respectively. Then the equation ofthe time-evolution, i.e., the density matrix equations forthe system can be described as dρdt = − i ~ [ H, ρ ] + Λ ρ inEq.(2), where Λ ρ represents the irreversible decay partof the E r ion system. i ˙ ρ = − Ω b ρ + Ω b ρ − Ω e ρ + Ω e ρ − iγ ρ + iγ ρ ,i ˙ ρ = − Ω b ρ + Ω b ρ − i ( γ + γ ) ρ , (2) i ˙ ρ = − Ω e ( ρ − ρ ) + Ω b ρ − (∆ e + i γ i Γ) ρ ,i ˙ ρ = Ω b ρ − Ω e ρ − [(∆ e + ∆ b ) + i γ i Γ] ρ ,i ˙ ρ = − Ω b ( ρ − ρ ) − Ω e ρ − (∆ b + i γ + γ + γ ρ , where ρ ij = ρ ∗ ji (i,j=1,2,3) and the density matrix elementswere constrained by the conditions: ρ + ρ + ρ =1.Here, γ ij designates the decay rates from | i i to | j i .In the classical electromagnetic theory, the electric po-larizability is a rank 2 tensor and defined by its Fouriertransform ~P e ( ω e ) = ǫ α e ( ω e ) ~E ( ω e ), which can be cal-culated by the trace computation of the definition ~P e =Tr { ˆ ρ~d } = ρ d +c.c.. Here, in the E r -dopped opticalfiber we consider the polarizability of the incident field ~E e carrying out at the frequency ω e . Therefore, we adoptthe explicit ω e dependence α e ( ω P ) ≡ α e , and ~E e was setto parallel to the atomic dipole ~d so α e as to be a scalar.Then its expression can be represented as follows: α e = ~d ρ ǫ ~E e = | d | ρ ǫ ~ Ω e , (3)The classical magnetic polarizations in the E r -dopped optical fiber can be achieved in the same way,i.e., ~P b ( ω b )= µ α b ~B ( ω b ), which can be obtained by themean value of the atomic dipole moment operator via ~P b =Tr { ˆ ρ~µ } = ρ µ + c.c . For the simplification, wechoose magnetic dipole is perpendicular to the inducedelectric dipole in accordance with the classical Maxwell’selectromagnetic wave-vector relation. Then the magne-tization α m is scalar, and its expression is as follows: α m = µ ~µ ρ ~B = µ | µ | ρ ~ Ω B . (4)The relative permittivity and relative permeability ofthe E r -dopped optical fiber can be given according tothe Clausius-Mossotti relations considering the local ef-fect in dense medium[21, 22] as follows: ǫ r = 1 + N α e − N α e , µ r = 1 + N γ m − N γ m . (5)In the above, the expressions for the electric permittiv-ity and magnetic permeability of this E r -dopped opticalfiber were obtained. In the section that follows, we willdiscuss its left-handedness via the permittivity, perme-ability and refractive index. III. RESULTS AND DISCUSSION
With the steady solutions of Eq.(2) for ρ and ρ , wecan show the numerical results for the electric permit-tivity ε r , magnetic permeability µ r and refractive indexn in the E r -dopped optical fiber. Before the calcula-tion, some typical parameters should be selected. In thefollowing numerical calculations, all the parameters willbe scaled by γ = 90.6 s − [23], and we choose the de-cay rates as γ =1.19 γ and γ =0.31 γ from Ref.[23].And we choose the average density for the E r ions asN ≈ . × m − , the electric transition dipole momentfrom | i ↔ | i is chosen as d = 2 . × . × − C · m [24] and the typical magnetic transition dipole momentis chosen as µ = 7 . × − Cm s − [25, 26]. The Rabifrequency of the electric component of the probe laserfield is set as Ω e =0.5 γ with ∆ b =0.25 γ . FIG. 2. Real (solid lines) and imaginary (dashed lines) partsof permittivity ε r as a function of the rescaled detuning pa-rameter ∆ e /γ for Γ=1 . γ , Γ=1 . γ , Γ=1 . γ , Γ=1 . γ . According to the refraction definition of the left-handed material n = −√ ε r µ r [27], we plot the permit-tivity ε r , permeability µ r and refractive index n versus∆ e /γ with different incoherent pumping frequencies Γin Fig.2, 3 and 4. The coincide intervals for negative Re [ ε r ] and Re [ µ r ] will demonstrate the left-handednessin the E r -dopped optical fiber. In Fig.2, we noticethe value and interval for negative Re [ ε r ] are increas-ing when the incoherent pumping frequencies are var- ied by Γ=1 . γ , Γ=1 . γ , Γ=1 . γ , Γ=1 . γ . Whichdemonstrates the incoherent pump field coupling level | i into upper level | i can incur the creasing negative Re [ ε r ].On the same parametric condition µ r was plot in Fig.3.It’s noted that Re [ µ r ] maintains negative value in the in-terval of [-15 γ , 0] when the incoherent pumping fieldmodulates its frequency, which shows the E r -doppedoptical fiber has negative Re [ µ r ] in the same intervals asnegative Re [ ε r ], and demonstrates the left-handedness inthe E r -dopped optical fiber. FIG. 3. Real (solid lines) and imaginary (dashed lines) partsof permeability µ r as a function of the rescaled detuning pa-rameter ∆ e /γ for Γ=1 . γ , Γ=1 . γ , Γ=1 . γ , Γ=1 . γ ,and the other parameters are same as in Fig.2. The plots for Re [ n ] in Fig.4 also demonstrate this.Fig.4 shows that Re [ n ] maintain negative value in theinterval of [-15 γ , 0] and its values are gradually enlarg-ing when the incoherent pumping field is modulated inthe same way as Fig.2.The reason may come from the growing incoherentpumping field, which drives the transition | i ↔ | i andbrings out the changing populations between | i and | i in the E r -dopped optical fiber. The variation of popula-tion between | i and | i results in the field-induced inter-ference effect on the electric and magnetic polarization,which leads to the increasing left-handedness eventually.Above all, the incoherent pumping field plays a impor-tant role in implementing left-handedness and attractsus mostly. What’s the result for the strongly incoher-ent pumping on the transition | i ↔ | i ? In the follow-ing, the incoherent pumping rate was set as Γ=10 γ ,Γ=15 γ , Γ=20 γ , Γ=25 γ in Fig.5, 6 and 7, which aremore stronger than those utilized before. As shown inFig.5, 6, we note that the intervals for simultaneous neg-ative Re [ ε r ] and Re [ µ r ] are [0 , − FIG. 4. Real (solid lines) and imaginary (dashed lines) partsof refractive index n as a function of the rescaled detuning pa-rameter ∆ e /γ for Γ=1 . γ , Γ=1 . γ , Γ=1 . γ , Γ=1 . γ ,and the other parameters are same as in Fig.2. when the incoherent pumping field drives | i ↔ | i heav-ily. However, the increasing pumping rate Γ leads to thedamping values for Re [ ε r ] and Re [ n ] in Fig.5 and Fig.7,even so the slightly rising Re [ ε r ] in Fig.6. As demon-strates the decreasing left-handedness when the incoher-ent pumping field drives | i ↔ | i heavily. It means thestrong incoherent pumping field plays a destructive rolein implementing left-handedness in the E r -dopped op-tical fiber. FIG. 5. Real (solid lines) and imaginary (dashed lines)parts of permittivity ε r as a function of the rescaled detuningparameter ∆ e /γ for Γ=10 γ , Γ=15 γ , Γ=20 γ , Γ=25 γ ,and the other parameters are same as in Fig.2. From Fig.4 and Fig.7, we conclude that the different in-tensities of the incoherent pumping field result in the in-creasing or damping left-handedness in the E r -doppedoptical fiber. The reason may qualitatively explain as the E r ion’s closed-loop configuration. When the incoher-ent pumping field drives transition | i ↔ | i heavily, itinfluences the interferences between | i ↔ | i and | i ↔ FIG. 6. Real (solid lines) and imaginary (dashed lines)parts of permeability µ r as a function of the rescaled detuningparameter ∆ e /γ for Γ=10 γ , Γ=15 γ , Γ=20 γ , Γ=25 γ ,and the other parameters are same as in Fig.2. | i simultaneously. The diminished quantum coherenceand interference decrease the left-handedness and lead tothe shrinking of Re [ n ] in the E r -dopped optical fiber. FIG. 7. Real (solid lines) and imaginary (dashed lines) partsof refractive index n as a function of the rescaled detuningparameter ∆ e /γ for Γ=10 γ , Γ=15 γ , Γ=20 γ , Γ=25 γ ,and the other parameters are same as in Fig.2. In our scheme, we used a three-level atomic systemto model the emission transitions I / − I / and I / − I / in an E r -dopped ZBLAFN optical fiber.In order to achieve an asymptotic simulation result,the decay rates were set from some relevant researchresults[23]. Undoubtedly, these parameters utilized thesimulating process promote the probability in the comingexperiment.From the above analysis, the left-handedness wasdemonstrated in an E r -dopped ZBLAFN optical fibermodeled by a three-level quantum system. In the litera-ture, there was a quantum optical method, i.e., photonicresonant materials[26] to realize left-handed media. Inwhich a controllable manipulation of the left-handednessby using the external fields (e.g.,just like the incoher-ent pumping field in our scheme) interacting with co-herent atomic vapour. However, in the current work, weachieved the left-handedness in the solid candidate whichis the prominant feature other than the coherent atomicvapour. We would like to point out one can fruit thecandidate of E r -dopped ZBLAFN optical fiber for left-handedness in our scheme. In the relevant experiment,the the solid candidate may be liable to left-handedness.Meanwhile, the candidate for left-handedness of E r -dopped ZBLAFN optical fiber paved a solid candidatefor left-handedness and extended the application of therare-earth-ion-doped optical fiber in metamaterials viathe external incoherent pumping field, which may at-tracted some interesting in the future. IV. CONCLUSION
In conclusion, our scheme achieved left-handedness inan E r -dopped ZBLAFN optical fiber via the incoher-ent pumping field driving the transition of I / − I / .Through simulating E r ion system with a three-levelquantum system, a gradually growing left-handednesswas achieved by the faint growth of the incoherent pump-ing field. However, the damping left-handedness emergeswhen the incoherent pumping field drives the transitionstrongly. Our scheme may propose a new avenue for im-plementing left-handedness and may extend the applieddomain for rare-earth-ion-doped optical fiber in metama-terials. ACKNOWLEDGMENTS
This work is supported by the National Natural Sci-ence Foundation of China ( Grant Nos. 61205205 and6156508508 ), the General Program of Yunnan AppliedBasic Research Project, China ( Grant No. 2016FB009) and the Foundation for Personnel training projects ofYunnan Province, China ( Grant No. KKSY201207068). [1] P. Urquhart,
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